Bad connections anywhere can cause electrical problems.

Since almost every electrical circuit in your car is routed through one of the fuse boxes, of course it can cause electrical issues.

Something else to consider, on some makes since the early 2000's, a fuse box isn't just a fuse box.

Often one of them, or both  (engine compartment or passenger compartment ((or any other compartment))) is an Integrated Power Module.

Basically a Body Control Module. It's a computer. It receives Voltage Signals from your controls (headlight switch, wiper/ wiper speed, tail lights, A/C controls, etc.

It's on one of the CAN Bus networks. 

The part you see up top with the fuses and relays is just the "thin icing on a 3 layer cake".

So, considering this, heck yes, a bad "fuse box" can cause all kinds of electrical issues from components not working to your car behaving like it's possessed by the devil. (wipers, headlights, A/C, etc turning on/off by themselves.

Yes it can. And if your car is ever flooded this is a big problem area because moisture gets trapped in there. Even people who drive around without the lid on have problems with fuse boxes.