Based on our experience, I would say that it is definitely possible. We moved here after plans for a 3-month backpacking trip fell through and I'm so glad we did. We've done a mix of trips and will have seen quite a bit of Europe by the time we return home in 2008 (we moved here in August 2006). Just to give you an idea as to where we've been, I've copied the list from our blog (including places we've booked for 2008). Our budget is 10,000 pounds per year for 2 people, but we typically stay in hostels or do camping tours. We also eat almost all our meals from grocery stores. I definitely second Vicky's recommendation to see the UK, that's all we could afford to do when we first arrived and I'm so glad we did! Oh, also if it helps I get 29 days of vacation and we always plan trips during x-mas, easter, and bank holiday weekends to take advantage of having those days off.
2008 Travels
Netherlands - Amsterdam
Germany - Berlin
Czech Republic - Prague
Austria - Vienna
Hungary - Budapest
Romania - Bucharest, Brasov, Cluj Napoca
Bulgaria - Black Sea Coast
Turkey - Gallipoli, Istanbul
Greece - Athens, Meteora, Kavalla
France - Brest, Dinan, Mont St. Michel, Bayeux
2007 Travels
Portugal - Lisbon, Sintra, Cascais
Spain - Barcelona, Madrid, Toledo
Ireland - Galway, Aran Islands, Kinvara, Dingle Peninsula, Ring of Kerry, Killarney, Kenmare, Kinsale, Cork
Belgium - Brussels, Bruges, Gent
Monaco - Monte Carlo
France - Calais, Paris, Avignon, Antibes, Nice
Scotland - Edinburgh, Stirling
Italy - Milan, Venice, Bellagio, Como, Cinque Terra, Pisa, Florence, Rome, Vatican City
Wales - Tintern, Blaenaforn, Brecon Beacons, Monmouth, Snowdonia
England - London, Stonehenge, Salisbury, Wells, Glastonbury, Bath, Stratford-upon-Avon, Coniston, Coventry, Nottingham, Avebury, Winchester, Gloucester, Portsmouth, Dover
Switzerland - Zurich
Germany - Frankfurt
Egypt - Cairo, Luxor, Aswan, Abu Simble, Alexandria
2006 Travels
Wales - Cardiff
England - Warwick, Ironbridge, Oxford, London, York