I don't even think the so-called budget airlines are that cheap, and in my experience are almost always more expensive when you want to take a suitcase.
Recent examples of trips I've looked at in the past year where a non-budget airline has been better:
Dublin -> London = CityJet were offering a flight with baggage for £21.90 (plus I got a £5 off voucher from Opodo for "Liking" their page on Facebook). I didn't even need the baggage on this occasion, but it was still cheaper than the starting price of the budget airlines for the time I was travelling.
London -> Berlin (and back) = British Airways holidays offered a package deal with return flight including baggage, and 3 nights in a hotel, for £515 for the three of us. Again, I didn't need the baggage on this occasion, yet it was still significantly cheaper when the hotel was factored in.
Glasgow -> London = British Airways offered a later flight time for almost the same price as one of the budget carriers.
Bristol -> Dublin = Ryanair had a cheap flight, but when baggage was added on, took them over the price of Aer Lingus' mid-range option that includes baggage. In the end decided to travel with Avios from London.
Plus, if you collect Avios, which is easy to do with a good credit card, BA is almost always cheaper, as single flights within Europe, as well as domestic flights are £17.50, including baggage.