Are AP scores fairly accurate? How can we check?
I'm not sure what you mean by “accurate." Are you worried that your exam was not graded properly or merely that the score doesn't reflect your true ability?
If you suspect that there might have been an error in the scoring of multiple choice section, you can order a rescoring of your answer sheet. See this link to order yours. This service costs $30 per exam, must be requested by October 31 (in the fall after the test was given) and can take up two months (!) to be completed.
You can also request a copy of your “Free-Response" booklet. This costs only $10 per exam (and is only available for the most recent exam administration), but it does NOT include any comments or grades. There is no re-scoring service offered for free responses nor is there any avenue offered for appealing your score. You can find additional information about this service and its deadlines here.
If you suspect scoring mistakes, you might want to spend the 30 bucks on the Multiple-Choice Rescore Service, even if the odds are slim that errors were actually made. But, given what college costs these days, the price of a pizza or two could bring you some peace of mind … or perhaps even some “I told you so" satisfaction.