Whereas /u/RestrictedData probably has something to say about the perception that the Soviets "copied" the atomic bomb, I once wrote a reply arguing that Buran wasn't a "copy" of the Space Shuttle:
Tu-4 really was a reverse-engineered copy of B-29 though, as far as I know.
I don't know about 'exact copies', but 'taking concepts' has happened from time to time in the US Military. Take for example, the Infantry Fighting Vehicle. First commonly identified as the BMP-1 in the 1960s, the US had no equivalent. (There is an argument that the Germans had an IFV before the Soviets, with the Schutzenpanzer Lang), but it was the fact that the Soviets had started fielding the things which really got the US's attention. Today, we have Bradleys.
A more egregious case, though, is that of the M551 Sheridan. Originally, the US Army had developed a light tank named the T92. The Army wanted an air-droppable tank of small size. T92 met this requirement very handily. Prototypes were built (one still exists), and the Army is about ready to put it into production.
Then the PT-76 is released by the Soviets. Basically a boat with tracks and a small gun, this recon vehicle, sometimes viewed as a light tank, had no direct counterpart in the US. Congress sees this thing, and asks the Army "Say, that new light tank we gave you millions for, does it swim?" "What, the T92? No, why?" "The new Soviet light tank. It swims" "So? We never wanted ours to swim" "But theirs does" "Bully for the Soviets" "But we are supposed to have better equipment. We give you lots of money. Why doesn't T92 swim?" "We don't want it to." "Look, if we're giving you all this money, you're telling us you can't build a tank which can do what the Soviet tanks can?" "We didn't say that." "Great. So in that case, if you want any more money from us for a light tank, make it swim".
(The above is, obviously humorous paraphrase).
The bottom line, though, is that work on T92 was abandoned, and the M551 Sheridan was produced instead. Like T92, it could be air-dropped. However, unlike T92, it could swim. It was also made with an aluminium hull to keep it light and floaty which made it vulnerable in Vietnam, was never particularly successful (Though the gun was fantastic), and the vast majority of the things were removed from service very quickly, painted up to look like Russians, and were used as training vehicles in Fort Irwin.
This just made the front page so you probably saw it, but apparently the U.S. got copies of Soviet maps and the State Department uses them.