'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 (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. 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'.

Before we discuss how bullet-proof armour was, some background on how armourers went about making 'armour of proof'. Armour can be made stronger in two ways - by making the metal stronger, or by making it thicker. Steel with sufficient carbon content can be hardened by heat treatment - heating it and then plunging it into liquid. This can be a cold liquid (as is the case in a 'full' quench) or a hot one (as is the case in a 'slack' quench), or it can be briefly dipped, interrupting the quench. A full quench will over-harden the metal and leave it brittle, so that it must be re-heated (tempered) to bring it to the perfect combination of hardness and ductility to survive sudden shocks without breaking or bending too much. However, heat treatment requires high-quality steel to start with - it must have sufficient carbon content to harden. There is ample metallurgical evidence that the carbon content of armour decreases in the later 16th century and into the 17th century. 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.

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. It is better than nothing, but isn't the comprehensive protection offered by 15th century armour against the weapons of its own day.

Thus, armour 'Of Proof' worked against some attacks, particularly from lighter firearms - perhaps sometimes even against muskets, if it was of sufficient thickness and quality.

Sources:

Alan Williams - The Knight and the Blast Furnace

Matthias Pfaffenbichler - Armourers