The law made is illegal to remove a serial number in 1968, but no mention of guns that were made before 1968. The statement below when taken in context of the requirements of manufacturers post-1968 to place a serial number in a certain way that is "not susceptible of being readily obliterated, altered, or removed."
Gun Control Act of 1968, Sec. 101, 82 Stat. 1213. To accomplish this objective, section 178.92 requires that each manufacturer or importer utilize an individual serial number for each firearm manufactured or imported and prohibits the duplication of any serial number placed by the manufacturer or importer on any other firearm. Furthermore, section 922(k) of the GCA makes it unlawful for any person to transport, ship, possess, or receive, in interstate or foreign commerce, any firearm that has had the importer's or manufacturer's serial number removed, obliterated, or altered.
I used to be solidly in the camp of "if any gun has the serial number removed - it is illegal to own", but the GCA of 1968 was a forward looking act and did not state that firearms without serial numbers that existed before the act was passed is subject to all the rules of the GCA.
Let me add that I would still not own a gun without a serial number or one that had one obviously removed. Primary concern is I worry that some law enforcement officers are firmly in the camp of any gun without a serial number being illegal no matter how old it is.