The answer to this depends upon the firearm, the circumstances (range and angle of attack) and the breastplate. While almost any breastplate would resist certain forms of attack (sword slashes, for instance), firearms are much stronger than other late medieval weapons. As the late middle ages transition into the early modern period, stronger forms of gunpowder and more powerful guns give arquebusiers and musketeers even more firepower. Even in the 15th century firearms test the limits of contemporary armour's protective abilities.

First, some words about circumstances. Angles of attack are important any time we talk about armour penetration. Shallow angles of attack direct a blow's force less efficiently through the armour, making it more likely that the blow will just deflect off the surface. Armours use this to their advantage with a variety of design features - in the Middle Ages we see curved surfaces that offer very few right angles. In the Early modern period we see breastplates that present a strong central ridge in the center of the torso or even a king of shallow cone at the lower chest, presenting an angled surface when the wearer is attacked from the front. In general, glancing blows are less likely to penetrate.

Regarding the firearms, early-mid 15th century handguns (short metal tubes on sticks) were 'relatively' weak, but still capable of delivering between 500-1000 joules of energy, based on Alan Williams' experiments. By the end of the century the arquebus (which looks a great deal more like a gun, but is still fairly short barreled) offers around 1300 Joules of Energy. In the 16th century we see both the introduction of new types of guns (the long-barrelled, large-bore musket, originally a heavy gun fired from a rest) and stronger 'corned' gunpowder. This means that a late 16th century rest-fired musket is over 3 times more powerful than a mid-15th century handgun! However, keep in mind that these are -muzzle- energies, and that bullets would have a great deal less force at greater ranges.

Late medieval armour's metallurgical quality differs significantly from piece to piece, between low carbon steel (or occasionally merely wrought iron) armour to heat treated armour of medium-carbon steel. There is a correlation between the perceived quality of the armour (finish, the presence of armourer's marks) and the metallurgical quality, though not an absolute one. The armours of some famous armouring centers, like Milan or Augsburg, are of a consistently higher quality than the armours of other places.

In the 15th century, these differences could be the difference between life and death for the person wearing a breastplate. While a heat-treated Milanese armour with the mark of a famous armourer might resist a mid-15th century handgun (at least at moderate ranges and angles), an unmarked breastplate of low-carbon steel made for use by infantry might not.

This difference in quality was known at the time, and was checked by what today we would call 'destructive testing' - they shot at breastplates. To quote a previous answer

'Proofing' armours by shooting them is well attested from the mid-14th century onwards, and particularly in the 15th and 16th century. The armour would be shot by a projectile weapon at a certain range, and would pass the test if it was not penetrated. This armour was thus 'proofed' and was said to be 'armour of proof'. Originally the testing used strong crossbows, and shifted to using guns in the 16th century. Thus earlier armours do no have visible bullet marks but these become more common into the 16th century. Quality control of armour was difficult in the days before modern measurement tools, so testing the strength of armour by actually attacking it was a good way to make sure it would protect the wearer. Sometimes 'proofing' was done by the maker, sometimes by the client (in the 16th century Emperor Maximilian II took to personally shooting his breastplates with an arquebus to ensure that they would protect -him- from bullets), and sometimes by municipal or royal arsenals as armour became more mass-produced and standardized in the 16th and 17th centuries. Degrees of proof appear early - in the 14th century there is a distinction between fully and 'demi' proof armour. In the later 16th century armour was 'proofed' to different levels - some armour might be only pistol-proof, while the strongest (and heaviest) breastplates would be musket-proof. This is also the origin of the modern 'bullet-proof'.

It is critical to note that early on the difference between armour 'of proof' and other armours is its quality, not its thickness. This changes in the 16th and 17th centuries as guns get stronger - so strong that no 2mm sheet of steel will stop a musket ball, regardless of quality. In addition, the metallurgical quality of armour (or at least its carbon content) -declines- through the 16th century, until many 17th century armours are made of wrought iron (which is to say, iron with no carbon).To quote from a previous answer:

Alan Williams cites contemporary documents to suggest that one cause of this is the supply of cheap iron from blast furnaces that was 'fined' (decarburized) in finery forges and turned into low carbon steel or wrought iron. With lower carbon steel to work with, hardening armour was not possible, so the only thing to do was make it thicker. Secondly and more importantly the sheer strength of 16th century guns meant that heat-treating armour alone was insufficient to protect against weapons like muskets, so armour had to be made thicker to protect against them. This thicker armour was heavier, which was a contributing factor in the decline of full armour (along with changes in tactics, armoury composition, and military strategy), as explained by me here and in this answer by u/hborrgg. However partial armour was extensively used in the 17th century, into the era of the English Civil War - there was less armour on people but what they did wear was thicker. In the Musee de l'Armee in Paris there is an armour of Henry IV from around 1600 that covers only the torso upper arms and upper thighs - it weighs as much as a 15th century knight's full plate harness.

At the same time that we see this increasingly thick armour, we continue to see breastplates of around 2mm thick (a 'comfortable' thickness), particularly in armours for infantryman. Could the thicker armour 'of proof' protect against firearms? Yes, though which firearms depended upon its thickness. This was acknowledged - distinctions were drawn between 'pistol proof' and 'musket proof' armour. To quote from my previous answer again:

So was all this armour 'of proof' actually capable of stopping bullets? The proof marks themselves and documentary evidence suggest - yes, it was, within certain limited circumstances. Armour wasn't 'proofed' at a few paces, but at more moderate distances like 50 or 100 yards. Alan Williams made extensive calculations of the ability of armour to resist various attack in 'the knight and the blast furnace'. Making a back-of-the-envelop calculation based on his figures, if a musket bullet (with an energy of around 3100 J) struck a 4mm thick low-carbon steel breastplate (assuming early modern low carbon is only 75% as tough as modern mild steel, as Williams does) at a 45 degree angle it would fail to penetrate, but if it struck perpendicularly it would penetrate. If the armour was simply made of a poor quality wrought iron, however (like many were) then a musket firing at a decent range could penetrate it at almost any angle. However, some breastplates of the later 16th and 17th centuries are even thicker than this-6mm thick or more. These might indeed be 'musket proof' at long or medium ranges, even from favorable angles of attack, but they would also be massively heavy. Against lighter guns (pistols and calivers, or the earlier arquebus) these 'proof' armours would offer more certain protection - it is quite probably that the mark on this armour may have been made by a lighter gun such as a pistol. Even the 4mm thick breastplate would be twice as thick as the average breastplate of 200 years before, and according to the numbers above would not offer protection at all angles or ranges. It is also important to keep in mind that much of the armour of the late 16th and 17th century was only partial armour, so for the breastplate to save your life the bullet would have to hit you there first.

So in both eras we see that some armour would stop the contemporary bullet, and some would not. However, the armour 'of proof' in the 15th century is just high quality armour - not much thicker than other armours, a full harness (suit) of it could still be worn comfortably. Bulletproof 17th century armour is massively heavy, and so the full harness of armour becomes impractical, and even partial armours are used in a decreasing number of roles. Thick breastplates (proofed at least against pistols) live on among heavy cavalryman into the Napoleanic era and beyond. We also see siege or 'sappers' armours made to protect people operating in the trenches around a besieged fortress. As guns dominate the battlefield more and more, non-bulletproof armours lose much of their utility, and so we see infantry ceasing to wear armour as the battlefield becomes more and more dominated by firearms.