Is it fine to bunch together power wires?

You are better off having a jumbled mess than having all the cables neatly organized.

Neatly organised is good, touching is bad. My .5, 2" numbers work very well, for NOISE. As I stated..

You shouldn't tell people that.  This isn't a Cray supercomputer, where there is an actual reason for the wires being carefully jumbled as you stated... His system would CRC because it was digital. The errors were cause because of just what you stated. Field migration from HC lines within the box.  Whopper in war games... Right..

It matters very much for noise, AS I STATED. I don't care about the gobbledygook you spouted.  The magnetic field may be affected, BUT you or I can't hear the affects of it.  At 30 db or a 140 db, in my systems. Whether through XLRs, RCAs or Speaker interconnects.

I'm pretty sure, I'd be walking out on one of your installs, because of the floor noise. I said  IC running parallel. There is NO NOISE when at least 1/2" apart, and 2" with PC. If a PC cross and IC, 90 degree, right angle, again 2" apart. They will SOUND better. Typical lineman for the county, answer, typical electrician answer. 

LOL I have a family chockablock full to the top of Union electricians. Top notch men.. They all learned one day at a BBQ.. The best of the best..
They learned about Stereo gear, the ones that could still hear anyway.

They learned about cable direction, not one started out believing it, they all did before leaving.. One made up an RCA and we tested it. Sounded better one way than other.. The spool of cable was cooked on a cooker for 3 days. live and learn, they did...

Two of the 7 actually specialize in HiFi, High Current cable installs, NOW! Go figure, they alway come to Uncle Scotty with stereo questions.

Regards

Can XLR cables run bunched up together?

XLR cables have good common mode rejection (assuming the signal is really balanced and the receiving end is using a differential input or transformer to cancel out the common mode signals). This means that any induced signal from neighboring cables should be nearly the same on both signals within the cable, and therefore be cancelled out. 

So, to answer your question, it's much less likely to be an issue with balanced XLR cables than with other cables. Also, the current in balanced interconnects is fairly low (assuming reasonably high input impedance on the component you are connecting to), so the amount of coupling is likely to be low.

It's probably not optimal to bundle them tightly together, but recording studios will often run lots of balanced cables through the same conduit without issue.