You have either a shorted Motor or a direct short on the interconnecting wire. By disconnecting and re-connecting the fans you will find which fan is bad.

Follow-up:

Thanks, Hollywood, but I have disconnected all the fan wires (at least from the pull switches) and still have a short.

I have now started taking apart all the outlets and wall switches.

Isn't there a better way to diagnose this?

Thanks again.

If you know the order of things on the circuit, you can be most efficient by breaking the circuit in the middle. Then cut that section in half, etc. until the source of the problem pinpointed.
It might be helpful to have someone in the bedroom and bathroom to watch and listen carefully when a fuse is replaced. They might be able to hear something or even see a puff of smoke before the fuse goes.

RPXLPX is right..... Before undertaking a major task, let;s determine if the short is from wire to wire or shorted to ground. At the breaker pannel, disconnect white and black wires. Ohm out between black and white and if you have continuity, 2 wires are touching. Ohm black and white to ground to see if it is shorted to ground. See if this gives you any idea where the short may lie....

Hope this helps.

IF you still have the fan motors wired in and their swithes on then you will not be able to determine if you truly have a direct short by using hollywoods method. Your multi tester will read continuity through the motor coil if the motors are not disconected. However it will work just fine if you have disconnected all motors from the circuit.

Thanks again.

I first disconnected all the fans (undid the pigtails at the fan switches).

I still had the short - 3.5 ohms by shorting an outlet on the same circuit.

So I tore apart the outlets and switches and it looks like the short is not there (I left them disconnected).

Now the fans show the same 3.5 ohms across the black and white wires.

Silly as it sounds, I am considering taking the fans down one at a time, cuz I don't know how they are wired - there are no large gauge wires going into the switch. They must go somewhere...

The fans are 40-50 year old Hunter Originals.

I haven't disconnected the fuse block at the box yet, but I did take the covers off and everything looks good in the box.

Is that the next logical step, and/or where do the heavy solid copper wires connect to the fan?

The heavier copper wire is in the electrical box in the ceiling above the fan, provided that they are installed in an electrical box. I have come across many ceiling fans which are mounted using a hook in a ceiling joist and the connections are just wired pigtailed to the fan in the attic. In one scenerio I had a fan mounted this way which spun on the hook twisting the wires around until the connection broke loose. This could be a possibility. If you have access to the attic I would begin searching for your short there. good luck.