I have an unusually high palate. Is that something that's bad?

No. Don't worry. High arched palate is a normal finding and is not a speech or eating problems.
Probably not an issue. A high palate is a normal anatomic variation. A high and narrow palate can be problematic. If your upper jaw is narrower than it should be than your lower jaw it can affect speech, swallowing, occlusion, and jaw function. Have the shape of your jaws checked by an orthodontist to assure that your jaws are proportional.
Maybe. People who have narrow face with high palate have large turbinates and as a result have problem breathing through the nose. People with marfan's syndrome also have high palate.