In the Heller decision, United States v. Miller was cited in part to show that the 2nd Amendment only applies to certain types of weapons. There are very different interpretations as to what exactly the Miller decision means for the 2nd Amendment, partly because it is seen as a test case initiated by the federal government in order to further New Deal gun control measures, but more so because it allows for a hybrid thoery to emerge with respect to collective versus individual rights theory.
In Parker v. District of Columbia, the Miller case was interpreted to mean that individuals have the right to posses and use weapons of the kind "in common use at the time", which is the source for the language to which you allude.
It remains an open question exactly what limits the government has regarding the type of firearm one may posses, but the Parker case would provide some solid footing with which to launch a challenge.
Interestingly and somewhat ironically given the current state of affairs, in Miller, the United States was arguing that the 2nd Amendment "protects only the ownership of military-type weapons appropriate for use in an organized militia" since they were tying to prosecute Miller for possession of a sawed off shotgun, which was prohibited under the NFA of 1934.
In other words, if the sawed off shotgun was in use by the military at the time, it would have been protected, it follows logically then that individuals would have a right to own any weapon which was also in use by the military, in addition to those that weren't, but which were legally permissible under the NFA.