Router

Assuming you are using a router and / or switch to connect the computers to the modem, the number of computers connected are limited by the number of ports that can be cascaded off of the router.

I know that I can connect at least 11 PCs to my router.

What does happen, is that the more PCs connected, the slower the internet connection is for each PC.

If there is a problem with cable modems burning up, it is only because the manufacture may not have designed it for 24/7 operation.

The important thing is to check both the modem and the power supply for the modem after several hours of "heavy" usage.
If either are hot, you need to install fans near them to keep them cool.
Or, purchase a "better" brand of modem.

Chas