The Smithsonian Air and Space Museum has an interesting article on the subject.

The long and short of it is: it's not that likely. The Soviet Space program used "Ivan Ivanovich" dummies before sending live cosmonauts into space, and the recovery of these dummies after reentry is most likely the ultimate source for the legend of Lost Cosmonauts. Apparently the workings of the space program were so secretive that recovery teams didn't know they were recovering dummies until they figured it out for themselves, so any bystanders would see recovery teams rushing to a capsule and pulling a lifeless body in a cosmonaut suit out.

The rumors gathered force, and grew more and more elaborate (and macabre) over time. The rumors were even repeated by members of the US Congress and by a columnist in the Washington Post up until the 1980s. Once glasnost happened in the late 80s, official records were opened up and the Lost Cosmonauts were proven to just be a legend (although there was evidence that the Soviet space program did cover up fatal accidents that occurred from fires and explosions on the ground).

Another aspect of the Ivan Ivanovich dummy tests that is interesting is that apparently the Soviet space program needed to play recordings from the space craft to ground control in order to make sure that the microphones and radio transmissions systems worked properly. They went out of their way to pick recordings that would not be considered from live cosmonauts (choir music and borscht recipes) in order to make sure anyone eavesdropping wouldn't think manned satellites were already in space. Apparently that part might have backfired.

One last irony to point out is that the Soviet space program was so cloaked in secrecy and rumor, that once Yagarin and Titov were sent into space, many people actually believed that this was a hoax, until President Kennedy confirmed that these were successful Soviet manned missions.

Source:

Ivan Ivanovich and the Persistent Lost Cosmonaut Theory. Smithsonian Air & Space Museum, 2017