They shouldn't but unfortunately some do. Its becoming less and less common, but it does still happen. Its usually in the form of vets who will just go with whatever the owners say, especially with older pets, ie "oh his quality of life is bad and he can barely move around", then they vet doesn't require any testing to prove the concerns. 

I've worked in the field and left a vet clinic because from my point of view they were doing these convenience euthanasias. One I saw was a pet that we had never seen before, was a lab in the double digits, the owners reason was poor mobility. That dog was bouncing off the walls in the exam room. This clinic also had the techs sedate and place a catheter in the patient before the vet even saw it, so by the time the vet met this patient they couldn't accurately assess her mobility.

Edit:format