This question may seem trivial amongst many people, that would instantly state "The decline was caused by the development of powerful firearms, that made the armour obsolete." as an answer to it, and so I would like to not only present arguments against that statement, but also go a little bit further and answer several other arguments that may come up during the discussion.
Now, concerning the effectiveness of the armour. I've gathered some information that clearly states that the production of bulletproof armour was not only possible, but actually existed in 15-17 centuries. I have brought several examples of such armour, the pictures of which you can find on the internet. Here are the examples of bulletproof breastplates of 16th - early 17th centuries with dents from bullets and musket proofmarks(first and the last one): http://imgur.com/a/cAaLa Not only that, I've also managed to find some book sources that can confirm that(I hope the authors don't mind me using the scans from their books): "The armourer and his craft" by Charles Ffoulkes and "The Knight and the Blast Fournace" by Alan Williams: http://imgur.com/a/eyPRj ; and here's a very interesting table from the Williams's book: http://i.imgur.com/sNTuna8.jpg that shows us the resistance needed to defeat a steel plate of varying thickness and material it's made of, under different angles and also energies that weapons of the discussed period could offer. It allows us to make a conclusion that you would need a ~3.5 mm thick steel plate of the best quality to get a protection against any weapon you could face in a battlefield of that time(except a cannon of course) or 4 mm thick steel plate of the lower grade steel, plus a good shape would only increase it's defensive capabilities.
But let's get back to the pictures of bulletproof breastplates of 17th centuries I've already shown you. You might notice that they don't really look like they are made of the best quality steel, and you'll be right: they weigh 15-17 pounds and have a thickness of 7-9 mm, which allows me to make a conclusion, considering the table we've just seen, that they were made from the cheapest material available, most likely simple iron. And no wonder, armourers of the period had two ways of increasing protection that armour could provide: by using better materials, or by making the armour simply thicker. Naturally, many preferred the second way, due to economical reasons. But just how expensive such a breastplate would be? Here's the link that provides us with the information(it also uses some of the sources above): http://medieval.ucdavis.edu/120D/Money.html
Armor of proof is tested during the making with blows or shots from the strongest weapons of the time; if a weapon is listed, the armor does not claim to be proof against everything, only that it is proof up to that weapon's strength (eg pistol proof is not musket proof, but may be sword proof)
Cuirass of proof with pauldrons 40s
Cuirass of pistol-proof with pauldrons 26s
Considering that musket's cost was 16s 6d - 18s 6d, that might be expensive for a common footman, still though, affordable, but let's take a look at the wages(late 16 cen):
1 shilling = 12 pence (d)
Captain 8s/day
Lieutenant 4s/day
Ensign 2s/day
Drummer or trumpeter 20d/day
cavalryman 18d/day
infantry 8d/day
And so it's obvious that it would take quite some time for a footman to gather enough money for a cuirass with pauldrons, but considering that we don't know what kind of material the cuirass was made, even if it's made of the cheapest material, he could still save a coin or two and get only a breastplate that would still greatly increase his chances of surviving a battle. But if you're of a higher rank, it should be alot easier to get a decent protection, so the cost wasn't really a reason for the decline as well, especially if you're of a noble blood. I would also like to mention the vastly increased volume of iron production in the 18th century due to the Industrial revolution, thus making the potential price for the armour even lower.
Was it the weight then? As was said before, a breastplate of proof would weigh 15-17 pounds, which isn't much at all for a soldier. Considering that it could be made from better materials and thus alot lighter, the weight wasn't a problem as well, at least not a bigger problem that the armour was in previous centuries, even if someone would like a bulletproof suit of armour(though I believe it would be costly and pretty heavy for a not well trained man).
So I hope you see now that this is not really a simple question. Taking everything that was said into account, one can easily imagine a picture of an 18th century line battle with infantrymen wearing breastplates and officers and cavalrymen in full harness, but as the history tells us, that never happened, and cuirasses only seldom appeared on cuirassiers and weren't even bulletproof. And that's why I'm here: to figure out either the real reason behind it, or the flaws in my arguments. Thank you for your attention.
You see a lot of armour still used by elite troops in the 17th century, such as Imperial Cuirassiers in the 30 years war, Polish Winged Hussars and so on.
Swedish cavalry did use moose leather undercoats and steel breastplates until 1707 or so, when the breastplate was no longer used. It was however, worn unde the blue overcoat, probably to protect it from rust.
Good quality armour was expensive as well as heavy. Unlike older armour, such as chainmail and lamellar armour, which can easily be modified or repaired in the field, you cannot modify or repair a breastplate for another soldier without a skilled smith and all his tools (and his forge).
Modern age armies became much larger than earlier armies, but still lived off the land. March dispersed and fight concentrated became the standard modus of operation. This, however, meant that troops marched back and forth quite a bit more, and were required to be able to do forced marches to join with other forces that had encountered the enemy. Considering the soldier was already carrying musket and bayonet, bullets and gunpowder, usually come cooking utensils, a backpack with some emergency rations, part of a tent and all his personal items, adding the weight of a cuirass would probably impact negatively on his ability to fight.
At the end, however, the soldier did not equip himself. He was equipped by his colonel raising his regiment or by the state conscripting him. For them, it quickly becomes a question of numbers. You can buy almost 3 muskets for the cost of a bullet-proof cuirass.
Do you want an army with cuirasses, or an army twice its size without? Considering that the vast majority of soldiers died of disease during this era, a unit with cuirasses would not have such a huge advantage over one without, especially not on a campaign. There's still plenty of unarmoured points which can knock the cuirass-equipped soldier out.
As for soldiers equipping themselves with armour - it would be unlikely that he would be allowed to keep it, as officers wanted uniform dress to be able to tell their units apart.
I'd like to make a couple of points.
First of all, firearms evolved and become better all the time. More powerful, with better rates of fire, cheaper and so more easily available in large quantities. So an armour that was sufficient to survive in the field of battle in the 17th century was not necessarily enough in the 18th or 19th.
The main changes were probably the introduction of line infantry, with the ability to put out massive firepower, and then in the second half of 19th century rapid fire rifles and the machinegun.
Then you have the fact that armour in some form was still in use in the 20th century. There were Cuirassier (armored cavalry, using a cuirass composed of breastplate and backplate ) regiments in the french, german and russian armies at the start of WWI. The french ones used the cuirass in the field for some weeks at the start of the hostilities.
(There are still some Cuirassier regiments today for ceremonial duties, like the Italian Corrazzieri.)
There's often a technological (and economic) determinist strain in answers to this question that disappoints me, the idea that the decline of armor was natural and inevitable due to objective efficiency gains. It doesn't surprise me that it was your studies of 16th century armor and the practice of bulletproofing that had lead to your suspicion of this approach, as that's where the narrative began to fall apart for myself as well.
The military-economic argument usually tries to point to large armies, cost, emphasis on mobility, state provision of equipment, and other factors related to centralized nation-states - and it seems quite convincing at first. I don't think this quite works, however, as we can observe many empires through human history with larger armies than the 17th century tended to see (eg. Rome) that showed no sign of abandoning armor in the same way.
On the other hand, the technological determinist argument usually asserts that firearms at some point became too effective - though exactly when and what is often pretty vague. The idea is that at some point, the power of the massed volleys became too great to make armor worth it. Again, there's something to this, but as you've read "The Knight and the Blast Furnace" you know that a bullet slows down dramatically after firing. Bulletproof plate was never meant to protect against extremely short range fire, but to push the range at which enemy fire can kill you to force arquebusiers to close the distance.
So, to some extent I'm as stumped as you are. The working answer I've come up with is a combination of these factors, in particular army size, state provision of equipment, and increasing numbers of firearms. Larger armies are composed of lower quality troops who lack the resources to supply themselves with armor and whose superiors lack the incentive to do the same, and this effect is pushed over the edge by the increasing number of firearms forcing the weight of armor ever upwards.
I'll admit, though, that I'm not satisfied with this answer.
It just had to wait until it caught up with the technology if weapons. We use personal armor at work.
