Yes. Downloading and playing games will increase power demands.
Unfortunately most mfg. will install the bare minimum size power supply, memory, and other parts in their machines. Also, power supplies tend to weaken with age and heavy use. It is possible that the psu has weakened and may be causing this problem. Another issue may be heat build-up inside the case. You can remove the side panel of the case {for better ventilation) while operating the computer to see if this makes a difference.
If you tell more about your machine, I'm sure you will get many recommendations as to how large a psu you should have as a replacement.
I recently posted about my computer shutting itself off without warning or error messages. Sometimes my children can play a game for 15 minutes before it turns off, sometimes 30 seconds. The same goes for me and downloading things like pictures. When I'm surfing through the internet, it hardly ever shuts off. I've turned off alot of my startup programs and downloaded the newest drivers for my system, but the problem persists. My question is does the computer pull more power when downloading or when games are played where this could possibly be a power supply going bad? All the fans are working, but I've had this computer almost a year and I've been reading how the power supply can start to give out. I've also tested every drive, the processor and the motherboard with the pc doctor that is installed on this computer and all the tests come back as a pass. I'm sorry this is so long, but I'd really appreciate any feedback. Thanks so much!

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