Absolutely not, anastasiav. I have the Nikon Coolpix 7900, and it can absolutely handle that shot. That being said, I have taken fuzzy pictures when conditions just haven't been good for a steady hand-held, and that does usually mean resetting it for a faster shot. It sounds like you've tried every setting -- including the sports setting which will be the fastest of the pre-sets (if not, try it). I doubt that something is wrong with your camera. Consider if you've had problems taking steady shots before (shaky hands?). Have someone else give it a try. And then, of course, if it's still a problem, go get yourself a cheapie little tripod and wired remote. I can't speak for the 5900, but the 7900, basicly your camera with a few more features, has excellent low light potential. Stunning, really, so this shouldn't spell the end of your use for this camera.

Consider if you've had problems taking steady shots before (shaky hands?).

I sat the camera on top of my car for both these shots, and (as you can see) it didn't seem to help. But I don't think camera movement was the issue.

and that does usually mean resetting it for a faster shot

I confess this is not something it occured to me to try. I started with the "regular" setting and went through slower and slower settings in hopes of having a long enough exposre to get a good shot. I never went faster.

I'll try faster next time....

No, no, you want to go faster if camera shake is the problem. Though balancing on a car should give you a good result, unless:
i) you're still pressing the button with your finger. no good, you need a wired or wireless remote;
ii) you have it precariously perched and wind is rattling it;
iii) there is some sort of vibration interfering with this, say of subways or the like.

Good luck!

Post script: of course, a faster exposure means less light, but this is much less of a problem in the digital realm...

It could also have been that there was too much other ambient light around.

There's definitely movement of the camera (car?) in the shots. I'm not sure if the wired remote is an option as the camera doesn't have a "bulb" option, but I might be confusing a couple things.

If the faster speed still shows blur, you can use the "self timer" option after setting the camera on a steady surface to release the shutter without moving the camera. Although, I think a speed setting faster than normal might be too much.

1. Camera on steady surface/tri-pod
2. Self Timer
3. Experiment with different speed settings.

Your images look cool; I like how the clouds light up.

Thank you, everyone, for your kind suggestions and help.

Self-timer will help a lot, in my experience. Also, go for real solid things on which to rest the camera (walls, etc.). Also try zooming out a bit, if you can. Night telephoto shots are about the lowest light levels you'll encounter.

I suggest, predictably, that what your pictures need is a little nudity.

But "nudity" to me is a pretty broad umbrella, so sexy clouds might count. I don't know.

NYT Circuits Digital Photography (special section from June 7, 2005).

Civil_Disobedient had a great answer a few days ago in AxMe about setting up a digital camera for different shots (MeFi is down right now so I can't verify that he answers your question but I do think it will be helpful).