Why does the instant coffee that I store in little plastic tubs go bad?

In a word, humidity.

Instant coffee absorbs moisture from air rather well. I suspect that your container isn't perfect airtight, and it's in a humid place. It's possible that the container is airtight, but enough water vapour gets in each time you open it to cause problems. This will be more of an issue for small amounts in a big container.

To test and possibly address this, try a small, definitely airtight container, such as a glass jar with a screw top. Even plastic boxes with an O-ring seal and clips aren't always well-designed; especially if the lid had been hot it may not actually seal perfectly.

You can also use silica gel sachets (food grade versions are available) in the jar with the coffee. These don't last forever, but they can be dried in an oven and reused, also they're cheap. Don't forget to store the spares tightly sealed.

Of course, if you're spooning some out with a damp spoon that will get water in. I have seen people sprinkle the coffee directly into a mug of steaming water - and the steam collects in the container. Don't do that - if you want to sprinkle it into your mug, do so before adding hot water.