Quite often they can, it all depends on how old the bird is and how big the lump is. A well looked after budgie may easily live for 12 years and often longer. There are various reasons for lumps; a common one being cancer, where it can be removed by a vet providing it is not to bad. Small lumps can also form from 'scaley-face', a condition where a minute mite infests the legs and cere (nose). Scaley-face can easily be cleared up with a paint-on solution available from your pet store.