I have a pair of Bose speakers attached to my PC. They’ve been fine for years but suddenly developed a buzz. I’ve checked that all the cables are firmly set. The same buzz is heard through headphones.
How do I tell whether this is a speaker issue or something internal to the computer?
I’m not going to be a lot of help, but are you plugging the headphones into the speakers? (I have seen speakers with a head-phone jack). If so, have you tried connecting the headphones directly to the PC (assuming the jack is the same)? If the buzz goes away, implies a speaker problem of some sort.
I had a problem with buzz that also seemed to crop up out of nowhere. I got myself one of these things - iFi iDefender3.0 External USB Ground Loop Eliminator - and it was fixed. SO glad I found it!
How are the speakers connected to the PC - USB port or headphone jack? Are the headphones connected through the same port?
I have Bose speakers for my TV. IIRC, when the same thing happened to me, I cleaned all the connections with 0000 steel wool, sprayed them with WD-40, and then wiped/let them dry.
For me, it worked, and the buzz hasn’t come back in the subsequent year.
ETA: if you can, try plugging the speakers into another PC/laptop/source. This will help you narrow down the origin of the problem (ie, to speakers or to the PC).
Okay. I did miss
So/but I would try to spray contact cleaner into the jacks on the PC, too (in addition to cleaning the jack at the end of the speaker cable).
There. Think I’ve got it now 
There are an enormous number of computer/audio/anything_electronic problems that can be solved by cleaning connector contacts. Once upon a time, long long ago, in a place far far away if you had to walk there, I discovered you were looked upon with the same awestruck admiration for fixing someone’s disk drive if you had to uncover an excessive propagation delay randomly causing the controller to miss sector starts or if you cleaned the connector tabs on the back of the drive with a pencil eraser.
I plug the headphones into the speakers normally. I just crawled under the desk and plugged the jack directly into the computer and got the same buzz. I’m doubting that both the headphones and speaker jacks became suddenly dirty in the same way at the same time.
If the problem is inside the computer, where is it likely to be and how hard would it be to diagnose and fix? Should I just bring it into a shop since I don’t know nothing about no computer insides?
This is a little USB adapter that lets you connect the headphones via a USB port, avoiding the speaker jacks. It’s eight bucks, so it might worth trying to see if the buzz continues. Also, can you plug the headphones in another computer or device? Do you still get a buzz?
If we are reaching, the two one-of-a-kind occurrences of this that I’ve seen:
My speakers and headphones have been in the same spots for years.
a buzz in speakers is almost always a ground problem. It could be that the plug strip you’re using is going bad or the electricity in your home is suddenly dirty. Have you recently added a new piece of electronics?
You can use a ground lift to isolate the problem.
Are the speakers powered through the USB or are they powered separately?
In the latter case, buzzes often accompany bad transformers. Try replacing the power cable/transformer. You can pick up new generic ones for $10-15.
I know you said this in response to @Noelq’s comments, but even if they’ve been in the same place I had a similar issue when one of the power bearing cords behind the monitor slipped into contact with the speaker cables. So, don’t dismiss it out of hand even if there was no movement of the equipment.
Having said this your comment that you’ve had the same issue with headphones plugged into the main PC jack entirely separate from the speakers makes it unlikely.
You could check the drivers for your sound system as a pre-bringing into the shop check…I had issues with sound being weird until I figured out that after the most recent update to Soundblaster it got stuck in surround sound mode.