Is it possible that a guitar could produce a low hum after replacing a pickup? I own a guitar that I (poorly) replaced a pickup on when I first started playing, and I was wondering if there was some sort of grounding that I could have messed up that could cause the hum? When plugged into my amp, I can notice the noise difference between two guitars, where an unmodified guitar is pretty silent, while my FUBAR solder job one produces a low hum unless I hold the metal shielding at the end of my guitar cable. The replaced pickup is a DiMarzio PAF-7 on an Ibanez RG7321. Any thoughts?
Yup, it sounds like you didn't get your grounding correct. When you touch metal on the pickups or the jack and hum goes away that's a pretty good confirmation.
I'd recommend letting a repairman look at it and fix things. You don't want to run risk of shocking yourself because of a badly grounded guitar. People have died from it.
Hum is produced in a couple of ways, and can indeed be related to a cold solder joint. If the hum goes away when you touch the grounding jacket of your guitar cable or a pickup case, then a cold solder joint or incomplete grounding is likely the problem.
To fix this, you are going to need to localize the issue by identifying the cold joint. One way you can do this is to employ a small alligator jumper and use it to bridge connections at the bare wire to the grounding lug on your output jack. Using this method, here's how I would find the cold joint:
First and foremost visually verify that everything is indeed grounded, and make sure that every component has a single path to ground. This includes pickups and potentiometers. One problem I often see on guitars with DIY wiring jobs is that the users often accidentally create a ground loop--which can create hum or even mute the guitar entirely--or they miss a component and wind up with incomplete grounding.
Once you have verified that the ground wiring is conceptually correct, using your alligator clip systematically connect the bare wire before each solder joint in the ground path of your wiring directly to the ground lug on your guitar's output jack. Pay special attention to joints soldered on the back assembly of volume and tone potentiometers--most ground leads converge there because it's an easy place to make a connection of two or more wires. It could be very different in your application however--especially since you rewired everything by yourself.
I usually test each joint with the guitar plugged in to a cheap-o practice amp at the lowest possible volume levels, that way I can listen for the hum and dynamically troubleshoot the problem. If you hear the hum go away after connection, then the lug associated to the bare wire you are testing likely has a cold solder joint. You can also look at the joint and tell if it's soldered correctly. A good joint should be clean and shiny, not dull, grey, or globular.
Another less interactive but more scientific way of fixing the problem is to fire up a voltmeter/multimeter. Most decent models have a mode that will measure if the two points you are touching are connected (a diode test). You can use this instead of an alligator clip and achieve the same results, but instead of audibly verifying that the hum has gone away at each connection you can scientifically prove that something is likely wrong with the path to ground between the two points you are measuring. It works (especially for assembling patch cables :D), but I prefer instant gratification when I'm hum busting.
If you make it through every joint in the ground path and still can't kill the hum then I'd drop by a guitar tech and get them to give it a look see.