For those of you who believe many cases are overturned during the appellate process, I invite you to subscribe to Find Law's Daily Opinion Summaries. I receive up to several notifications a day, and rarely does anyone include a reversal. Reversals simply are the exception, not the rule.
Often, appeals are denied because they are "untimely." The appellate process has various deadlines involved, and missing one of those deadlines can be fatal to that person's appellate process. The only remedy is to file a claim of ineffective assistance of appellate counsel, but many are excluded from this because they have filed their appeals pro se.
I received a summary just today regarding a Stay of execution that was awarded by the United States District Court for the Eastern District of Arkansas. Don William Davis was scheduled to be executed on July 5, 2006, by lethal injection. The stay was requested on the grounds the appellant needed time to litigate the constitutionality of Arkansas' lethal injection. The District Court granted the stay so the matter could be resolved.
The U.S. Eight Circuit Court of Appeals reversed the District Court's order on the grounds that the motion for the Stay was not timely filed. The Circuit Court found that the appellant had several years in which he knew that his execution was imminent and during which he could have filed his claim that the execution mode was unconstitutional without the need for staying a scheduled execution.