Well, you do have to care if you want to remain safe. First of all, the tight spots could include a weakened area in the metal. If the structural integrity has been compromised, it may not be safe to shoot.
Also, even if it is JUST a tight spot, you have to consider than when the bullet is being swaged down to make it through there, pressures may spike. And, some bullets, such as homogenous metal offerings, may not readily swage down and could actually cause quite accident. On the other end of the spectrum, extremely soft projectiles could leave an obstruction in the bore and wreak havoc on the ensuing shot attempt.
Until the extent of the tight spot is further explored, it pays to be safe instead of sorry...