The simplest answer to your question is that even with the best of early modern metallurgy, a musket-ball proof breastplate can't be made without making it thick enough to be excessively heavy. That said, some very particular types of soldiers continued wearing armour for many years, into the Napoleanic period and beyond.

The decline of armour is a very long process that does not finish until relatively recently. While armour in the 19th and 20th century is not my forte, the period between when heavy cavalry stops wearing breastplates and when infantry starts wearing body armour is relatively small.

When (practical and effective) personal firearms were first introduced in the 15th century, they could penetrate weaker armour, but high quality armour could be hardened with heat treatment enough to resist gunfire from most ranges (the same heat treatment would protect against couched lances, pollaxes and other powerful attacks). Heat treatment had the great advantage of making armour stronger without increasing the weight. And indeed, into the 16th century armour retains its thickness (around an average thickness of 2mm for the breastplate) and weight (a total full-body harness weight of 45 lbs, not including supplemental mail). High quality armour that could resist powerful attacks was 'of proof' - it was tested for quality in some way, sometimes by shooting it (around 1500 or so guns replaced crossbows for this purpose). With this protection, a man at arms in say 1480 was well protected against most attacks against him, both from missile weapons and in close combat. Keep in mind, though, that not all armour was heat treated - Italian armour without armourer's marks is generally not heat treated, and there are few surviving heat treated armours not made in southern Germany, Northern Italy or the 16th century Greenwich workshop in England. That being said, most surviving 15th century armour is southern german or north Italian, so armour from elsewhere is too small a sample size from which to draw a definitive conclusion.

However the 16th century saw three developments that made armour, at least full armour, less appealing. First, the development of better gunpowder ('corned' powder, which ignited with more force and consistency) and more powerful guns (long-barrelled guns fired from a rest - true muskets) made the most powerful mid 16th century guns more than 3 times as powerful as their early 15th century predecessors. Guns were now powerful enough to penetrate even hardened armour reliably at a variety of ranges. However, keep in mind that not all guns on later 16th century battlefields were full muskets - lighter arquebuses and calivers were often used, and cavalry was armed with pistols. In the 17th century, guns got more powerful and lighter, such that a handheld flintlock musket of 1700 equalled the power of a heavy, rest-fired musket of 100 years previously.

Secondly, tactical developments meant that more and more gunners were being deployed, and they were used more effectively in combination with pikemen. Where much less than half Swiss Mercenaries or Landsknechts would be armed with firearms in 1500, by 1589 the English expedition to the Normandy was 60% pike and 40% gunners. In the early 17th century, shot came to outnumber pike by nearly 2 to 1 in the English army and others (indeed, the Neatherlands under Mauriace of Naussau was a leader in developing firearms tactics), and different methods of deployment increased the volume of fire, until the introduction of the bayonet and the abandonment of the pike altogether by around 1700 or so.

These two factors meant that the battlefield in 1550 confronted soldiers with an order of magnitude more lead flying around than the battlefield of 1500 (let alone 1450). Armourers made some adaptations to the threat of firearms - breastplates from the mid 16th century to 16th present angled surfaces when viewed from the front, to aid in deflecting bullets coming from that direction. But in order to defend against more and stronger guns, the only thing to do was make armour thicker. In the 16th century (beginning, interestingly, around the time that firearms contributed decisively to battles at Biocca and Pavia) breastplates become significantly thicker. There are some breastplates that are still made at around the 'standard' of 2mm thick, but there is a a growth of super-thick breastplates - 4mm or even as much as 7mm average thickness. These are the bullet-proof breastplates of the later 16th century. The problem with them is that they are enormously heavy. Later 16th century armours without any supplemental protection and equipment weigh around 60 lbs, which approaches the maximum that a man of normal size can comfortably wear for an extended period, even on horseback. Late 16th century writers describe men at arms overburdened with armour and chaos as valets arm men at the last minute before battle (because the heavy armour was too tiring to wear for significant periods beforehand). So the increase in armour to the core of the body was made up for by dispensing with armour for the limbs. By the mid 17th century, only the breastplate, helmet and a gauntlet for the bridle hand remained of a cavalryman's plate armour - the rest of the horseman's body was protected by a leather buff coat. In terms of protection, the general trend is that all-around protection of the sort that armour provided in 1480 isn't possible in 1580, while at the same time armour (at least full armor) is heavier than it was before. Armour can defend against close combat attacks, but it can't defend against muskets at close range. If a soldier is wearing only partial armour, they are now much more vulnerable everywhere they are unarmoured.

This dovetails with a third development - the decline of the man at arms in favor of lighter forms of heavy cavalry. The thickening of armour in response to firearms is only part of this story - lighter cavalry (called Reiters or demilancers - either armed with a pistol or lighter lance) was cheaper to equip and more versatile. While men at arms were used decisively as late as the battle of Ravenna in 1512 and effectively as late as the battle of Marignano several years later (and used much later in the 16th century, as at the Battle of Ceresole), by the second half of the second century lighter forms of cavalry had replaced them - men at arms were both more vulnerable than they had once been, and more expensive compared with their alternatives. The cost of a man at arm's armour and horse barding remains steady while lower quality armour for lighter horsemen gets cheaper and cheaper in the 16th century, and the pay of these soldiers reflected this. Thus both the 'set up cost' born by an individual soldier buying his arms and the cost to a prince to pay them was different. Thinking about armour in terms of costs and benefits, the costs of armour were now comparatively higher and the benefits were lower (since against a heavy musket no one was as safe as they had been in their armour when faced only with longbows and lances...or even arquebuses).

Continuing to speak of cavalrymen, some portion of European heavy cavalry continued to wear a cuirass (proof against pistols, though less good against muskets at short range) into the Napoleanic period and beyond. Wearing armour conferred an advantage to heavy cavalrymen engaging other cavalry using swords and pistols (whose attacks the breastplate protected against). Our Napoleanic and early modern warfare flairs could explain more of this and the transition away from men at arms and toward lighter cavalry.

Regarding infantry, infantry armour probably peaks sometime in the 16th century and declines in use thereafter. Moving on foot, infantrymen were more burdened by heavy armour. It should be noted that armour was never universally used by infantry in any case - certain types of troops appear to have preferred to be unarmoured, and other were lightly armoured. However by the end of the 16th century musketeers were generally unarmoured (where Charles the Bold had required that his arquebusiers be armed with breastplates in the 1470's), while some pikemen retained their armour into the 17th century (presumably to protect themselves in the push of pike - armour protects against pike better than bullets). Infantry armour is effectively gone by the time pikemen are replaced by musket/bayonet armed infantry towards the end of the 17th century - as guns came to dominate infantry combat more and more, armour that would protect against swords or pikes but not guns was less useful.

Armour in the early modern era increasingly offered a marginal advantage in specific situations. It did not offer all-around protection against the full array of battlefield threats. In the middle ages by contrast, armour had provided just that kind of reliable protection against a wide variety of attacks. If you are going to wear something that weighs 45 pounds, it had better be worth it. In 1700 armour's cost and inconvenience was not worth the limited protection it provided in specific circumstances, except for a number of heavy cavalrymen, who continued to use it into the 19th century.

Sources:

Williams, Alan - The Knight and the Blast Furnace - this book provided much of the technical detail for this post, as well as general information about guns and their deployment

Ralph Moffat - The Importance of Being Harnest: Armour, Heraldry and Recognition in the Mêlée

European Armour 1066-1700 - Claude Blair

EDIT: added more detail about the commonality of heat treated armour and fixed some figures

As a follow-up if it was impossible at the time to properly create or maintain musket-proof armor, why weren't bayonets considered to be of sufficient danger to require armor? Wouldn't be advantageous for when melee combat came to at least have some units equipped with armor?