This is why you heard stories of people getting heavily underwater when the stock that had vested crashed. Not many of these stories recently, but the time will come again.
... you will never go underwater from holding onto a vested stock grant. (This is the situation the OP was asking about.)
... you will never go underwater from holding onto a vested & exercised stock option if you paid taxes out of pocket.
... you may go underwater if you take on debt to finance the tax liability incurred when exercising stock options and the value of the retained stock subsequently drops below the unpaid portion of the debt you undertook to exercise it. (This is the scenario that led to the stories you refer to.)
You can get in real trouble with options of non-public companies and with ESPP plans. Stock from an ESPP plan, if sold in the first 12 or 18 months of holding (depending on the plan details) will be reported as ordinary income at the time of purchase. If you sell during that 12-18 month period at a loss you will still owe taxes at the full price, and that can get you in a hole. I've heard of people owing 20k taxes on a 5k return because of a panic sell during a market downturn.
I'm just an engineer and not a tax professional, so there may be errors in what I've stated.
When you exercise them, the company's most recent valuation will include a value for the common stock you purchased, and you can get hit by AMT on the value increase from the strike price to the current value.
Note: I am not a tax expert and my memory is imperfect.