Would it not be possible to construct bullet-proof shields of acceptable weight and cost?
If it would have been possible and conferred a tangible battlefield advantage to the inventor, it would have swiftly become widespread.
The fact that we don't have such shields in history implies to me that it was 1) impossible, 2) too costly, and/or 3) not enough of an advantage.
For instance, like Nemowork's said, there are examples of 'bullet-proof' cuirasses. So you definitely could make a buckler-type shield that is thick enough to turn a bullet from a musket... But what then? It protects what, 10% of your body? Weighs enough to tire out your arm and to be slow in melee. And prevents you from using a musket or a pike. Not to mention probably costs a lot more than a normal buckler, since it needs to be so thick and from fine steel.
A cuirass would be better, like Nemowork's said, but much more expensive, probably more than most soldiers could afford. Cheaper and thinner cuirasses were still good enough to turn a pike or a sword, but a musket would punch through. (I'd probably still want one, though, if I had to be on the battlefield. Just on the off chance that it would deform and slow down the bullet enough to make the wound survivable... Assuming I could afford it.) Again, the fact that those cuirasses got phased out with the pikemen around 1700 (save for cuirassier cavalry) speaks of the military reality that they were not good enough (for the price?) against the contemporary muskets.
And as a final note... if you are imagining a 'shield wall' or a 'tortoise formation' of slow shield-bearers inching closer, I am sure that the opposing artillery is having a field day blasting them to bits.