Is there a limit to the amount of addons/mods I can have installed in Garry's Mod?
Yes there is somewhat, not a limit of what content or anything however there is a limit on how many Lua files may be loaded at once.
The current limit seems to be 4096. I doubled it so it should now be 8192. If that didn't change anything let me know, maybe there's another hardcoded limit within the engine and it might be a bit more tricky to remove.
So you can have as many addons so long as the number of Lua files does not exceed 8192.
I'm sure that THEORETICALLY, there's no limit or restrictions, but realistically, you can have the best computer and the world, and still not be able to have to many for the simple issue of conflicts. There will come a time where you have so many addons installed that installing any other addons will cause critical conflicts. It's like Jenga, but instead of removing blocks, it's installing addons. Sooner or later the tower's going to fall.