[skookum]

No, this is not why you heard such stories. If you use the commonly accepted definition of 'underwater' as applied to financial dealings which is that you owe more for something than it is worth, then....

... 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.)