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Question: if a router is sitting on top of a monitor (which I told them specifically not to do), and it heats up a lot, can that cause problems?

Yes, absolutely, without question, it will cause problems. It could, in fact, be the source of your reported behavior.

I had a wireless router system at our company once, for a short time while we were split between two buildings. Everything worked fine during the morning hours and during cloudy days, but as soon as the sun came out and it got warm in the afternoons, like clockwork, the wireless connection would start acting up.

One of the routers was stored in an attic, you see, and the attic would heat up to about 110 degrees F. Moving the router into a normal room-temperature area solved the problem instantly.
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Tony Fabris