I didn't see the programme, but I do think it is possible to tell when an animal is in pain. One of my cats sits in a particular position if she is not well - ie if she has a tummy upset or feels uncomfortable, I can also tell by the condition of her fur which deteriorates immediately if she is ill. She has congestive heart failure and renal failure so she is special needs in a sense that she must take pills and have a special diet. But she seems quite happy and just carries on as normal. Mind you, if I thought she was suffering in any way, I would have her put to sleep. I do think that if you know your pet, you can guage when they are suffering.
Not sure about a rabbit in a wheelchair though!