The cells eventually give up, but the life span is apparently longer than many had feared for the most recent generations. I don't have the link, but there have been some online torture tests where SSDs were brutalized and kept on chugging.
For what it's worth, my 3 year old Intel SSD has had 4,771 GB of writes and the Intel Toolbox says it has 99% of its lifespan left--not that I put much stock in that computation. It's well under half full, so I'm wondering if I'd ever replace it.
They can drop dead like anything else, but as far as I can see, that's less likely to happen than with mechanical drives.