Solder paste definitely goes bad over time.
Obviously nothing in the solder itself can happen, but the flux that is mixed in looses its potency over time, making the paste perform a lot worse.
If this is just some plumbers solder paste you use for soldering together some copper piping, that stuff will last forever. The flux there is much more aggressive and you don't need that fine of a solder joint, as long as it mostly fills the joint it will hold pressure.
But when you are talking about solder paste used to solder tiny SMD components on a PCB, you do NOT want to use old flux. The stuff does actually start to perform way worse after a few years (or even a year if the container is regularly opened and used). It still solders but it leaves a whole pile of stray solder balls around and in between pins, potentially causing shorts. It doesn't wet nearly as well so you get resistors floating up on top of a solder joint rather than grabbing it, resistors thombstoning etc. All in all it becomes a total shitshow.