The online web store provided by Google for Android is a charm and is something new for the Android user. However, a lot of improvement is currently is progress for the web store which might make the installation a easy game in the future. However, at present, the installation might take a time especially if you are on EDGE or slow wireless connection. The key to this problem is to have some patience and wait atleast 15 seconds for the installation to start or abort.
Another important thing you have to remember is your Android Marketplace account is connected with your Google Account and thus you can just check your web store account settings to get the recently purchased applications list and re-download the application again without paying for it again.
So in case your download or installation of a purchased application has failed, do remember to check the settings before trying to purchase it again. The problem has more to do with your internet connection that the Android Market Online. Also Android market is poised to become the best applications marketplace in future given the way Google is working into it.