Why are my nodes not showing up in a Panel?
Since the problem of non-displayed Panes can be caused by quite a lot of things, it can also take quite a lot of time to find a solution (2 days in my case). It turned out that I had overwritten panels-pane--node.php.tpl in my custom theme and that I accidently put an if statement around the whole code, which only returned TRUE for one specific node. Woops...

In order to make the searching process a bit easier for future users who have a similar problem, I will sum up some stuff to try underneath.

Maybe it's a theming thing

If you have a custom theme, your first reflex should probably be to take a look at it. (That being said, my reflexes aren't that good apparently.) First, use a tool like Firebug to inspect your Panel and make sure that the Pane isn't printed. Something like a lost visibility: hidden; can easily be solved in your CSS files. If your Pane just isn't there, maybe there is something wrong with one of your templates. Take a look at the templates that start with panels-pane.

Maybe it's a Panels thing

Take a look at the Pane's settings in your Panel. As CodeNext suggested in the comments above, check that the active style is No Style. In other cases, there can be a problem with (the CSS of) that style. Also, make sure that you didn't set a visibility rule that hides the Pane for you.

Maybe it's a clashing modules thing

P Magunia suggested that the jQuery version can affect the proper working of Panels. If you installed jQuery Update, you can temporarily deactivate this module to see whether something changes. In my case, nothing happened. Also, install the latest versions of all modules on your website. Maybe the Panels module clashes with another outdated module or is outdated itself (see  No Sssweat's comment).

Taking Jimmy Ko's remark into consideration, I reinstalled an old backup of my website, where the problem did not arise, and updated all installed modules. Since the problem was still there, I could convince myself that the hidden Pane was not related to any module. Yet another reason for backing up your website's (WAMP) folder and database.

PS: Maybe it's an alien thing

If none of the above hints worked, no error messages are shown, Google lets you down, and you have been clicking and typing around for days, probably some aliens hacked your website and are making fun of you. Beware, they are potentially dangerous. Never lose your temper...