"What could happen" depends a lot on the brand and model of bottom bracket and the riding and maintenance conditions it has been subjected. Different quality materials and design may have very different failure modes.
In my view, the failure of a particular BB may not be catastrophic, but may happen on a critical moment of the ride, resulting on an accident, injury or worst.
In the best case, the bearings will only develop some play. If that is the case, it first becomes annoying and ultimately causes problem with the front derailleur or chain guide (if any). In the worst case something may crack. The crack itself may not be terrible, but it may affect the bike frame if used in that condition.
The click you describe may be a crack or dent on one of the races, which may over time cause a bearing ball or roller to break, possibly causing the axle to bind.
This is what happened to me once: I had a Neco brand bottom bracket that developed a noticeable play (On a mountain bike frame with a mountain triple Shimano crankset). The resulting wobble would not cause trouble with shifting and at that time the bike was used for very short urban commuting on flat streets. I neglected for too long until one bearing cage snapped. The bearing balls fell out of place and the axle was bounced all over the shell.
The bike was rendered useless in a second. I could not accelerate or even stand on the pedals while riding a busy street. Luckily it happened right after a red light so no cars where buzzing past me.
The axle wobble in this bottom bracket caused te seals to be less effective, letting water and dirt in, which in turn corroded the bearing cage, accelerating the decay of the whole thing.