Another possibility—and one that ended up being the culprit in my particular case—is that your machine is making local backups when it's unable to connect to your external backup source. In this case, you can just disable the backups by entering the following command in the Terminal:

sudo tmutil disablelocal

Later, if and when you decide to re-enable the backup, you can reverse it with the following Terminal command:

sudo tmutil enablelocal

Another possibility—and one that ended up being the culprit in my particular case—is that your machine is making local backups when it's unable to connect to your external backup source. In this case, you can just disable the backups by entering the following command in the Terminal:

sudo tmutil disablelocal

Later, if and when you decide to re-enable the backup, you can reverse it with the following Terminal command:

sudo tmutil enablelocal

Update for MacOS Sierra and Later

It seems like the disablelocal and enablelocal commands are no longer available. As an alternative you can delete local snapshots by doing the following:

1. Get a list of the local snapshots by running:

tmutil listlocalsnapshots

You should get some output like

com.apple.TimeMachine.2018-10-03-145245.

2. Delete a snapshot by running:

tmutil deletelocalsnapshots <date of snapshot>

So, for example, using the output from step 1, you would run:

tmutil deletelocalsnapshots 2018-10-03-145245

And you should get some output like

Deleted local snapshot '2018-10-03-145245'