There have been many mounted bow troops. And when the Crusaders went to attack the Muslims in the Middle East, their small bows were harassing weapons. They could barely penetrate the mail of the Crusaders. This was before the advent of plate. They used bowfire to wound horses.
The Mongols were successful because of numbers. There weren't many standing armies back in the days of the Mongols. United standing armies equal in size to the Mongol horde were pretty rare.
Okay Celtavian, you just need to stop this. It is becoming increasingly obvious that you don't know what you are talking about. Speaking as someone with a Bachelor's degree in History (with Medieval Europe and East Asia as my specialties), most of the facts you have been spouting have been completely wrong. Rather than tear them all apart one-by-one, I will focus on the Mongol issue and use it to point out some of the various issues covered in this thread.
The Mongol's certainly didn't win battles using sheer numbers. After all, their biggest conquest was China, and it is almost a historical fact that no-one outnumbers the Chinese army. The Chinese weren't slouches either. The 13th century, Song-dynasty Chinese had pretty sophisticated "mountain-pattern" scale armor and even could field gunpowder artillery. There are plenty of historical accounts of the Chinese throwing gunpowder bombs off their walls at the besieging Mongols. The Mongols still won in the end using their horse-mounted archers and their brilliant organization and tactics. Of course the Mongols took gunpowder weapons for themselves and utilized them against their enemies, including the Chinese.
Throughout history, the nomadic tribes of the Central Asian steppes, who primarily relied on mounted archers, have repeatedly conquered everyone else in Eurasia. Their fighting ability can not be understated.
Anyways, guns and gunpowder have a much older history than people usually give them credit. Here is a rough timeline of the advancement of gunpowder weaponry (and the armor used to combat it):
10th-11th century: Gunpowder is invented in China and utilized in primitive incendiary and smoke bombs, as well as in fire spears and fire arrows. (In Europe, William the Conqueror invades England, Chainmail is still the best armor available, and the knight's lance hasn't been invented yet.)
12th century: Cannons and grenades are first utilized in warfare. (Richard the Lionhearted is off on the Crusades.)
13th century: Rocketry is first developed, and rockets are deployed as weapons. Cannons first spread to the Arabic world. (Later crusades, Mongols reach Europe.)
14th century: Ming China fields entire military divisions of gunpowder-using troops. The hand-cannon is developed. Gunpowder weapons reach Europe. Plate armor is developed and is worn by both mounted knights and foot-soldiers. (Hundred years war is in full swing.)
15th century: The matchlock arquebus is invented and widely adopted. Advancements with infantry trained with long pikes obsoletes mounted knights. Battles switch to armies of foot-soldiers wearing plate armor and armed with guns and spears. (The Battle of Agincourt is in 1415, Colombus discovers the Americas in 1492.)
16th century: Wheellock firearms are developed. Mounted archers are still being used by the Mughuls to conquer India (alongside muskets and cannons). Cortez conquers the Aztecs while wearing plate armor and carrying both a musket and a sword. Nobunaga Oda uses guns and swords to begin the unification of Japan. (Queen Elizabeth is alive in late 16th century.)
17th century: Around 1650, the flintlock musket is developed, finally creating a weapon that obsoletes plate armor. (The Scientific Revolution takes place.)
18th century: Flintlock muskets and field artillery fully rise to prominence, but cavalry swords and bayonet charges still play a major role in combat. (American Revolution and French Revolution occur towards the end).
19th century: The widespread adoption of rifling in gun barrels for the first time. The development, in rabid succession, of the minie ball, the percussion cap, smokeless gunpowder, metal cartridge ammunition, and repeating firearms paves the way for modern weapons. The Gatling gun and Maxim machine gun are invented.
20th century: Modern automatic firearms are invented and come into wide use. New materials are invented that can resist gunfire, making armor once again an essential part of the battlefield.
For some reason, a lot of people seem to block out large sections of this timeline from their thinking. When "early firearms" are brought up, they think of the 18th century, and completely forget about the seven hundred years of much more complicated history that preceded it. For example, Napoleon's troops were not using anything even resembling what could be called "early firearms", yet he seems to be continuously brought up in these discussions as if he did. It is also worth pointing out that plate armor and the knight are not directly correlated; plate armor only came into use in the twilight days of the knight, and lived on for centuries longer.
My background as a student of history makes me much more inclined to like having guns in D&D.