My boyfriend and I spent 3 weeks in Europe last August. We went to Dublin, Paris, Rome, Venice, Milan, Florence, and Gothenberg, all for under $4000 each, including flights. (We also went to London, but were lucky enough to be staying with a friend, otherwise the budget would have been much higher)

The three major expenses in a Europe vacation are:
food
shelter
transportation

If you plan well enough in advance, you can find great deals on transportation and shelter. We stayed in very nice hostels every night (save one night on a campground in Florence), and paid roughly 35 Euros a night for the two of us.
We took advantage of some great RyanAir deals, including a 0.01 fare from Milan to London, so including the airport fees, we spent 30 Euros on the flight for both of us.

Food can be VERY pricey, but there are ways around it that are very similar to the things you'd do at home. Things like eating a big lunch, getting take-out from restaurants, visiting markets, bakeries, and grocery stores, etc. will cut down on your food expense a fair bit.

And a note for anyone looking at getting a EuroPass for the trains around Europe - a lot of trains still charge reservation fees, even if you have a Pass. The passes are great for people who are winging their trip, but if you plan to be on specific trains, don't go this route. We would have saved money had we bought individual tickets, versus buying the pass and reserving trains in advanced.