Does anyone actually die because they're a titan?

I can't quite remember whether this is the case or not, been a while since I've watched the show, but it's stated that Titan shifters die after 13(?) years, and so they have to be eaten before they die to carry on their powers, but does this even happen? What if that's not even the case and they're just eating their former comrades without gain, and they could've been shifters for longer? I guess nobody tested it because they didn't wanna lose their shifter, but it might be interesting if it's just a lie.