Righties.. I wanted to make sure I got this right.
Okay- you're in your computer and you place a document into your drop box folder. Your computer will then sync your dropbox folder on the computer with your remote dropbox - how long this takes depends on the file size and your connection speed. Once it has synched, you will be able to see the document on your iPad. The document is
not currently on your iPad at this stage. Tap on the document, and you will see the spinning wheel as your iPad fetches the remote document. Tap on the star at the top left. This will make the file a favourite and start the process of creating a local copy of the document on your iPad. At this point, tap on the double star icon (centre-ish on the screen). This lists all your favourite documents and their current status. Here you can see the file being downloaded to your iPad, you need to make sure this is complete
before quitting or moving away from Dropbox (correct for iOS3.#, likely to change when iOS4.2 is released). You now have a copy on your iPad you can then open in an app of choice with the bendy arrow icon on the far left. (PDF's can be sent to iBooks for instance).
I'm guessing you just tapped the favourites icon and didn't give it enough time to download fully before going offline (no criticism - easy enough to do as dropbox isn't very explicit about when it actually has created a local copy). Watching the favourites list while it downloads should prevent you from doing that.
Hope that helps