I've been trying to come up with a nightmare segment for one of my characters, but I'm not certain how to set the tone and atmosphere correctly. I want the nightmare to be about the character's oppression and feeling of incapability/inadequacy. The thing I'm worried about most, is the segment becoming too cartoonish and/or over-dramatic. Any advice on how to avoid it?
Nightmare segments are interesting to write, because you can be as creative as possible with them. You don't have to worry too much about it being too vague or too unrealistic. In my own personal experience, stress-related nightmares usually take the form of failing to do something simple that you've always been doing fine with in reality, and then all hell breaks loose. It's a common trope, sure, but it does happen. So don't be afraid to be over-dramatic, because it's a nightmare. Nightmares are meant to be over-dramatic because when we sleep and dream, our sense of logic is shut off, thus releasing the floodgates of our emotional state of being. There's no real way to avoid this, so go nuts.
I admit I haven't tried this in writing, but some of my worst nightmares involve me trying to do something simple, like walking, and being entirely incapable. I know I'm trying to walk, my dream-brain is sending the dream-signals to my dream-legs, but I simply cannot lift them. When I'm in this state, now that I think about it, I'm not so worried about "why can't I move?" as I am of "I can't do the thing!". Almost like I'm accepting the circumstance as already making sense, because dreams don't have logic, so instead I'm worried about the inability to do the thing. If my nightmare was of me trying to run and rescue someone, I would not say "Why can't I move my legs?" but instead "I can't get there because my legs don't move and now everything is impossible!". The only thing I would specifically caution you against is that you might want to have just enough logic in the dream that the reader doesn't know it's a nightmare until you want them to know. If you have this character's body decaying, or dead relatives appearing just to taunt them, or other gruesome impossibilities, the non-reality of it might be given away too quickly.
I actually think nightmare segments are a great way to add character depth and explore a character's inner thoughts, emotions, etc. Dreams are a very good, yet subtle way to convey that. I agree with both Becca and Zero, dreams don't have to be super realistic, but keep it so that the reader doesn't immediately know it's a nightmare. And the more creative you are, the better. I think another important part of writing nightmare segments is the aftermath of it. Usually after nightmares, depending on the severity and what it was about, people are shaken up, anxious, paranoid. It's important to convey that to show both the readers and the other characters in the story that something really has gripped your character, and once again, could be a great segue into character development.
Read the replies above and they definitely seem to be putting you on the proper path. I'd say an important aspect of a good nightmare scene is the slow build that creates that suspense that sucks the reader in and then plays around with them. If you nail that, you should be effective!
It might help to look up certain elements that appear in dreams that are triggered by specific stressers. If there's something specific that's giving your character these nightmares, you could incorporate common elements that tend to actually appear in dreams like that.