There are two problems I see right in front of me with GAOTD: The lack of a real mirror link, and the download server not having enough bandwidth.
1. Mirror link. If you look on the download page, you'll see a big button for downloading and a mirror link underneath it. However, the mirror link redirects towards the file pointed in the main link, which means that there really isn't a mirror. If you look on the feed service, under download_link and download_mirror1, the links are exactly the same. So in case the main download server is down or getting overloaded, there's no way that you'd be able to download the giveaway except to wait for a rerun.
2. Bandwidth problem. As evident on April 8, the giveaway was a huge 241 MB backup software. If you downloaded the giveaway on that day, you might have noticed that the download was particularly slow. That seems to be because the download server was designed for many downloads of small 500 KB to 20 MB files, not many downloads of 50+ MB files. In fact, it was even difficult to get a connection to the server. I tried at least 50 times (real number, including tries my download manager made) before I could download the giveaway. This, combined with the lack of a proper mirror link, creates a great problem for people trying to download large giveaways in one day.
Suggestion: The bandwidth problem may be harder to solve, but the easiest thing to do right now is to load the giveaways on to some external uploading service. Whatever happened to Wikifortio and that other old service that GAOTD used to host their files on? If those were used, that'll provide perhaps double the bandwidth available for people to download the giveaways.