It has to do with training and the fact that that the range, accuracy, and deadliness of arrows and bolts are much overhyped.

You can take just about anyone and, with a few weeks of training, have a perfectly adequate musketeer. It takes years to do the same with an archer. At the same time, arrows are very unlikely to kill someone that they hit, and against even mild armor, have a tendency to not do much more than moderately inconvenience them. Not so the musket ball. Quoting a bit from a career Elizabethan soldier, Humfrey Barwick:

For in truth when I was in the French king's service among the old bands of footmen, I vigorously defended the force of the longbow, but this is how I was answered: Non non Anglois, vstre cause est bien sale car dieu nous a donnes moyen de vous encountrer après, unautre sorte que en temps passé. No, no, Englishman, said he, your cause has become a bad one, for God has given us means to the means to meet with you in another way than was the case in the past. For now the weakest of us are able to give greater wounds, than the greatest and strongest archer you have.

When I replied, as Sir John Smith often does, that the number of arrows come so thick that it was like a hail, he answered that it was not to be feared like a weapon that kills where it strikes. For, said he, when I march directly towards them [the archers] and see them coming [the arrows], I stoop my head forward a little, and so my burgonet will save my face. And when I see the arrows fall on my head piece; or on my chest; or pouldrons; or vambraces: well after seeing that, and noticing that they do me no harm and have no force, I bravely advance forward to meet with the enemy.

...

It so happened that at the conclusion of the peace-treaty and the yielding of the city of Lieth, that I had occasion to meet with various French acquaintances of mine. Among them I met with a gentleman and an old soldier, who served in the same company of footmen as I had when I was in French service. His name was Monsieur de Sentan, whose valour and honesty I will vouch for. I learned to know him as we were both under the same corporal and lodged together, as well as dining and drinking together for a long time afterwards. Given the great courtesy and respect I received from him in France, I tried to be considerate to him at Lieth. After we had spent some time talking together on various matters, I asked two questions of him.

The first was how many of our soldiers were slain during our attack on the city. He told me that 448 were killed. I asked how he know that, he answered that the Governor of Lieth had ordered that the uppermost garment of the English soldiers should be removed and brought into the marketplace to be counted. This was done and there were found to be as many as mentioned earlier.

Secondly, remembering the words of the Frenchman I quoted earlier on, I asked Sentan how many were slain from arrows from the beginning of the siege onto that very day. He answered not a single one, aside from one, he said, who was shot between the shoulder and the bodie, and that with the heat of his body when the arrow was pulled out, the head got stuck and didn't come out in one piece. The surgeon was forced to cut the man to recover the arrowhead, and so the man was still recovering from the wound. From this and other accounts I'd heard up to that time and since, it does appear that great numbers have come to their deaths by arquebuses, but few or none through arrows or archers.

Quotes from the 8th discourse of "A Breefe Discourse, Concerning the Force and Effect of All Manuall Weapons of Fire and the Disability of the Long Bowe Or Archery, in Respect of Others of Greater Force Now in Vse" via a transcription into modern English: http://the-norseman.livejournal.com/13230.html

One thing I have noticed in questions is that people do not fully understand the method of fighting war during the black powder age. As a result if the 30 Years War (1618-1648), nations restricted the method of warfare to be more "scientific" as to ensure that bloodshed of the mentioned war does not happen again. So war changed and became limited, battles didn't have more than a hundred thousand men in total until the later half of the 18th century.

So, war was formal, everyone generally fought in a line with muskets. Cavalry was in a support role until the French Revolution and artillery wasn't used in mass until the same Revolution. The reason is due to technology, muskets were easy to train with but inaccurate. It could take two weeks to become an expert with a musket while it takes a life time to perfect a longbow or crossbow. Further, there's a matter of logistics, where everyone was using a musket, so it would be impractical to include a new weapon set in your supply lines.

Finally, they are more fickle to the weather. Wood changes shape due to the atmosphere, so you don't have the consistency of a musket acting the same way under most weather.

In general, muskets are easier to use.

Let's reword this just to get past a bias in the question:
For warfare in the 17th-19th century western world, did any side use archers of some kind against their opponents who were only using firearms and melee weapons?

Others have explained for land combat the dichotomy of warfare in this period of interest.
If you want the maritime perspective - the 16th century has evidence for the use of bowmen at sea. The warship Mary Rose not only document having arrows for them, but archaeologists found the said arrows still onboard. (See the book Before the Mast: Life and Death Aboard the Mary Rose since it's full of reports on the findings and meaning of things recovered from the wreck) But, in the 17th century, I've yet to see evidence for any notable use of archers on ships. As mentioned below, the use of a bow requires training for years. In the course of the 17th century, the French and English organization of men on ships changed the concept that soldiers would be completely separate from sailors. Instead, the "fighting sailor" (basically, a sailor who not just operates the ship, but also work on gun crews and participate in boarding actions) developed over the century. Seamanship is a consuming skill to learn itself, and then to add onto that archer would seem a bit much. Even crossbows would seem pointless since you would probably do equal or more damage as a period smoothbore musket (especially on a ship that pitches up and down and side to side).
Hope this adds to the understanding of warfare in this period.

Im not sure what your asking. By the 16th century, vitually every army in Europe had adopted gunpowder weaponry augmented by pike phalanxes. And by the 19th century, rifled muzzle-loaders, breech loaders, and artillery, had increased the range and accuracy of most weapons into something we would call "modern". After all, everyones favorite rifle, the Mosin-Nagant was designed in the 1890s and remained in frontline service until after World War Two.

One thing that no one has mentioned. The point of war is not merely to just kill the enemy, killing the enemy is only one small part of disabling the enemy, which is just one small part of forcing ones own will onto others. The wonder of muskets is that they are excellent at controlling other people. For example, a group of 10 soldiers with Muskets can move into a town of 500 and effectively subdue the inhabitants of that town. A group of say, 10 archers simply would not be able to do that. A musket allows me to point it at someone and command them on how to behave, and chances are they are going to comply, even if I'm say 20 yards away from them.

Even aside from the superior effect of muskets in subduing an enemy population, they still have a lot of value in the battlefield, since they can do essentially everything a bow does, just with less cost, training and variation.