Does downloading too many games slow down a device? I have over 15 games on my device. Some do occupy some memory; others not so much. I have enough leftover memory on the phone.

Is it true that this can slow down the device?
When you download and install a game on your device, the data is stored within internal storage. When you run the game, it loads all the assets into RAM, which is Random Access Memory. Accessing RAM is much faster than accessing internal memory, which allows games to load scenes, animations, and other things faster. The game should know exactly where the assets are in RAM with an memory address. It can jump right to that point and pull the information down.

However, games are not the only thing that uses RAM. The operating system, system applications, and other installed apps are also using RAM. The more RAM that is used, the less there is for new operations to occupy temporarily. If RAM gets too full, then it could cause problems.

Leaving applications running in the background leaves RAM occupied. After a while, the memory should automatically be cleaned up, especially if you start new applications that require that space in RAM.

Another culprit that could be slowing things down is cache. Cache is used to store app information, web site data, and other information that the app will use to speed up load times. If too much cache is used, then it could be slowed. Games can use a lot of cache since they need to load images faster. Clearing the cache occasionally is typically a good thing.

TL;DR

Yes, running many games can cause a slow down on your device. RAM that does not get cleared and having too much cache are the main culprits. RAM should resolve itself over time, though closing the app completely is a good idea just to make sure. Clearing cache every so often is a good idea as well, especially if you run many of the games often.