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"If they ever make a movie about the life of I-278, it should be played by Dustin Hoffman."
In February 2003, on misc.transport.road, Andrew Kirschner posted the "Sad, broken family of X-78's." All of I-78's 3-digit interstates (the "x78's") have egregious flaws, and the father interstate acquits himself none to well either. Archived at Google Groups for your reading pleasure.
If numbering plans in 1957 and 1958 had taken hold, the family of x78's would have hung their heads in Ohio. In 1957, I-78 was proposed to lead from Norwalk to Akron, on a direct route that was never built, and from there to Youngstown, where I-76 goes today. In 1958, I-78 was changed to use today's I-680 around Youngstown, overlap with I-80 to the west, then use today's I-480 into Cleveland.
I-178 (preliminary numbering) New York
On Aug. 22, 1958, Interstate 178 was New York's submitted proposal for a freeway from today's I-278, over the Manhattan Bridge, to the proposed Lower Manhattan Expressway (to be part of I-78). A week later, AASHO replied that I-178 should be I-478, as it connects two interstates. 
See also: Manhattan Bridge (Steve Anderson)
I-178 (cancelled) Pennsylvania
Never built, Interstate 178 was a planned connector from today's US 22 (formerly I-78) into Allentown. It was shown in Rand McNally atlases in the late 1960s, but was not included in the 1971 federal interstate route log. 
See also: Pennsylvania's Decommissioned Interstates: I-178 (Jeff Kitsko)
I-278 New York; New Jersey
35.62 miles; from US 1 in Linden, N. J. to I-95 in the Bronx. Interstate 278 never touches I-78, which terminates in Jersey City. Expressway and bridge names include the Staten Island Expwy, Verrazano Narrows Bridge, Goethals Bridge, Brooklyn-Queens Expwy, Triborough Bridge, and the Bruckner Expwy. I-87 ends at I-278.
Yes, it would have intersected I-78
Plans for a longer I-278 in New York and New Jersey were cancelled, leaving the route truncated at both ends. In the west, I-278 would have been extended to meet I-78 in Springfield, N.J., then continue west along NJ 24 to end at I-287. The highway would continue as NJ 24. This section was deleted around 1970. 
The abandoned I-278 west extension is why there is no ramp from northbound Garden State Parkway to I-78 west; it would have made more sense to take I-278. 
Sheridan extension proposed, cancelled
Surprisingly, the original 1958 proposal for I-278 in New York State is the highway's current location. . No other 3-digit interstate in the area survived without a number or location change.
In the interim, however, a northward extension using the Sheridan Expressway was proposed. When the short Sheridan opened in 1962, it was designated part of I-278; the remainder of I-278 leading eastward became I-878. .
I-278 would have continued along an extension of the Sheridan, moving north then east to meet I-95 near Baychester Avenue. This plan was cancelled by New York Gov. Rockefeller in 1971. 
Part of old 278 becomes I-895
"I Hated, hated, hated this interstate"
Ty Rogers gives the road two thumbs down:
"... has got to be the worst excuse for an interstate highway in the whole country. It is a series of different roads thrown together. Many entrance ramps have no acceleration lane at all and have "STOP" signs where they join the expressway... there is a sign saying "Exit 4- I-278 BQE etc." To continue on I-278, one has to exit onto a single lane ramp. All this when I-278 should be the mainline and the Grand Central Pkwy should be an exit of I-278. I know that there are many other examples around the country of having to exit in order to continue on the "mainline" but all of these problems, IMHO, collectively put I-278 in the hall of shame." 
Though some parts of I-278 seem to have been built centuries ago, the oldest part of I-278 is the Goethals Bridge, built in 1925. 
I-378 (decommissioned) Pennsylvania
Interstate 378 went south from I-78 (now US 22) into Bethlehem. It was decommissioned when the new I-78 opened to the south. It survives as the 3-mile PA 378 freeway (378 continues south as a surface street).
See also: Pennsylvania's Decommissioned Interstates: I-378 (Jeff Kitsko)
I-478 (not signed as interstate) New York
2.14 miles; Interstate 478 is the unpublicized numbering for a stretch of highway including the Brooklyn Battery Tunnel. It occasionally appears on road maps as well, but is not signposted. The state reasons that saying "To I-278" is more useful to motorists. 
The original plan for a longer I-478 is much more interesting.
I-178 and the Manhattan Bridge
On Aug. 22, 1958, the New York Department of Public Works submitted a multi-route proposal including an Interstate 178 expressway, leading from today's I-278, over the Manhattan Bridge, and connecting to the proposed Lower Manhattan Expressway (to be part of I-78). A week later, AASHO replied that I-178 should be called I-478, because it connected two interstates. 
In 1971 the Manhattan Bridge Expressway was cancelled, and I-478 was routed along the Brooklyn-Battery Tunnel instead. 
The Westway proposal would continue I-478 north along Manhattan's western shore, connecting with the Henry Hudson Parkway. This was effectively killed in the late 1980's.
There was an elevated highway along the west side, called NY 9A, dating back to the 1930s. As the decades passed, the road became more delapidated and was closed to trucks. In 1973, an asphalt truck illegally entered the road, and caused part of the road to collapse. The highway from Battery Park to 57th street was closed permanently.In the early 1980s, it was demolished in stages.  
There were studies starting in the 1960s called the "West Side Highway Project". In 1969, New York asked the federal government to include the West Side Highway in the Interstate system.
In 1974, after considerable controversy, the state and city reached an agreement on I-478. In January 1977, USDOT Secretary William Coleman gave federal approval for 90% Interstate funding for the mostly-underground Westway. In August 1981, final plans were announced to build the Westway. It would have been a 6-lane tunnel built in earthen fill. 
Steve Anderson writes:
"The former West Side Highway, which was to be the new Westway (I-478) before losing out to the cause of striped bass in 1988, is currently being reconstructed as an at-grade "urban boulevard", complete with park space and "designer" lighting. It will carry signs for NY 9A." 
and: "The I-478 designation actually DOES exist, but as a secret route. It consists only of the Brooklyn-Battery Tunnel and its approaches, and does not include the West Side Highway or the replacement arterial (the "Route 9A Reconstruction Project"). This finding was double-checked from FHWA Interstate Toll Route listings (February 1997, available on the FHWA web site) and NYSDOT route logs... I just received a letter last week from an engineer at NYSDOT District 11. [I-478] was truncated after the Westway proposal was defeated in September 1985. To prevent confusion among motorists, the I-478 designation is not posted. Originally, I said that the I-478 no longer existed. Now I'm eating crow!"
Or how about I-695 for this route
A 1971 map (courtesy Stephen Summers) shows I-695 routed along the proposed Westway instead. I-478 was routed along the Grand Central Parkway between I-278 and I-678. (The GCP today is not an interstate.) 
I-678 New York
Steve Anderson writes:
"The Whitestone Expwy. was originally built (as the Whitestone Parkway) in 1939. The Van Wyck Expwy. portion of 678 was constructed between 1948 and 1961. The original designation for 678 in the 1950's went from the Bronx-Whitestone bridge along the Whitestone Expwy. and then onto the Grand Central Parkway to its intersection with I-278, just before the Triborough Bridge. However, because of its low overpasses, the Grand Central Parkway could not meet interstate standards." 
Fanciful extension: all the way up the Hutchinson River, Merritt, and Wilbur Cross Parkways to I-84 in East Hartford, Connecticut.
I-878 (not signed as interstate) New York
0.70 miles ; Interstate 878 is a secret number for a portion of the Nassau Expressway near JFK International Airport. It's officially the nation's shortest interstate, but it is not signed.
Formerly of the Bronx
Originally, the section of I-278 between I-895 and I-95 in the Bronx was called Interstate 878.