I've always wondered, given the heavy commitment to gunpowder-based warfare from the late 1600s onwards and the ever-increasing possibility of having a volley of lead flying at you, why did soldiers stopped wearing a breastplate as part of their kit? I get that it made them slower/less mobile, but with the exception of the occasional bayonet charge, musket infantry didn't exactly have to move much anyway. Was it just a case of firearms becoming too powerful to be stopped by metal armor?
There are several reasons for this.
Armour is expensive. As armies grew, it became a factor of wether you wanted more troops or if you wanted armoured troops.
Campaigns became more mobile and covered larger territories as armes became larger. Remember that only a small minority of the time of a soldier was spent fighting in those days. Marching not only to get from point A to point B, but also to forage, plunder, patrol and do recoinnasance. And everywhere you went, you needed to lug around every piece of heavy armour you had, wether or not you were wearing it or not.
While armour that could stop pikepoints, rapiers and the odd pistol bullet was not that heavy, armour that could stop a musket ball at distances down to 30 meters needed to be at least 3mm but preferably 4mm thick of good quality steel - and it would still be penetrated at distances lower than 30 meters (which were common as salvoes were fired at short distances for max effect). Such a breastplate (from a 500x500x4mm plate) would weigh about 8 kg.
Armour was always subject to being penetrated, and if it was, shrapnel of armour would follow the bullet into the wound, severely complicating the wound and your chances of surviving it. At some point, armour becomes so heavy, expensive and adds enough risk if it is penetrated that you simply won't bother.
/u/vonadler listed some good factors, but these factors were just as prevalent in the 16th century as well. What changed in the 17th century?
Flintlocks.
They greatly improved the reliability of the musket and replaced the matchlock.
In the English civil war the infantry of the new model army had pikemen with breast plates and matchlock muskets. After this debacle and while Charles II was on the throne a young Duke of Marlborough had an affair with one of his mistresses (possibly), the Duke of Marlborough would go on to command unarmored infantry using flintlocks.
As a follow up question, what about a big thick shield for a small percent of the army? Its true the shield would be subject to all the same problems that armor had, but it seems easier to have one thicker piece just in the front. Also if only enough for the front rank were brought much less total metal would need to be bought/carried. Using my skills of totally guessing, I would guess that a moderately thick metal shield would stop most bullets before modern firearms.