Well i've never had the delight of suffering from any major illness so i've never first hand needed to see the NHS for serious treatment, but i'll give the case of my girlfriend to think about.
Pretty much last year this time, my girlfriend started getting abdominal pains that were getting more painful, she had already taken a few visits to doctors to see if they could figure out what was wrong, within a few months she had gotten into such pain she had to go to A&E to get some treatment. They diagnosed her with Pelvic Inflammatory Disease and threw her on IV antibiotics, after giving her a laproscopy. This basically means they couldn't figure out what was wrong with her (as PID could mean any number of things) and decided some antibiotics would help.
Fortunately, the problems did go away for a while, but they've been gradually returning (although the cause may or may not be the same) and from some instances the pain was enough to even wind her (i've witnessed it). She did start seeking doctors again around winter and lots of promises were made, she was put on waiting lists and booked up for blood tests and scans - months later. By this point, she's had blood tests (where up to a pint of blood was removed) and they've LOST her blood tests, making her have to go through them again, she's recently had a scan that has miraculously also gone missing, she was booked to see a senior doctor (she's only seen inexperienced juniors so far for a case where it's clear that the problem is not black or white) this month, but that's been moved back another November by the hospital, because you know, why not?
You know the best part? She still doesn't even know what's going on, an entire year later, she's neither received diagnosis or treatment and it's in an area of her body that potentially carries high risk for her future, with the last 'diagnosis' being PID it could affect the reproductive organs. This is also for a person that suffers from Bipolar, it's not going to be particularly easy going day in and day out not knowing what's going on with your body, and she's not even sure if she might be just imagining it and if she's just going crazy, she wonders if she'll ever have kids if things go on if they do...
It's mental how the NHS can just be so negligent as to let cases like this fly under their nose, and at the end of the day, they'll cite excuses like she's low priority, and people will be there to back the NHS. Sure it might be free, and that might seem noble, but in individual cases like these, can you be surprised why people are pissed off.