I am sorry you are having problems with the downloads of CSV files in Constant Contact. I would like to help find out what is causing the issue. Can you tell me which downloads are not completing, what are the files connected to (i.e. contacts, events)?
Thanks for this clarification! Because you are working with the registrants in an event there are two workarounds we can suggest. The first being to download your file in a different browser or even a private/incognito window. The second workaround you can try is clicking on "Reporting" tab of your event and selecting to export a full or custom report instead.
Where you able to download your report through either of these workarounds?