YANBU!
Myself and DH (and several others I know,) have had difficulty with the NHS appointments system.
As you say @RedLemonadeNTaytos if the appointment they send is inconvenient (and it often is,) you can't just ring and change it, you are sent another one (which is often inconvenient as well!)
Also, we have had an appointment for say 15th July, and it's OK, as we are off work or manage to get the day off, and then a week before, we get a letter saying 'unfortunately we have had to cancel your appointment, a new one will be sent out in the post!' 
AND my DH had an appointment on 10th June this year, and 5 days before it, we got the automated 'let us know if you will be attending' call, and DH pressed 1 for yes! 48 hours later, someone called from 'the bookings team' to say the 10th of June appointment has got to be cancelled.
DH said 'but this was booked 3 months ago, and I only confirmed I was coming the day before yesterday. What on earth has happened in the last 48 hours that means my appointment in 3 days has to be cancelled?!'
The guy on the phone said 'no idea. I have just been asked by 'the consultant' to cancel it. ' DH was fuming, He had got someone to cover his shift at work, and so he lost his 10 hours pay! He couldn't ask the person who covered his shift to just not do it anymore! It doesn't work like that!
DH reckoned the weather forecast predicting lots of sun and warmth the following week, meant the consultant had cancelled all his appointments. We are actually sick to fucking death of this shit now. These consultants act like they are the most important people in the world, and no-one is allowed to question them. Fuck everyone else and how they have been inconvenienced! 
Several years ago, DH had to attend a hospital 4 or 5 times, that's 30 miles away from us. On 3 occasions, they made his appointment for 7.45 to 8.00 a.m. The hospital is right across the other side of the city where the hospital is, and they made the appointment for 8.00 a.m! It's an hour and a quarter trip in the car, when it's not the rush hour!
We had to leave at 6.00-6.15 a.m. to avoid the bloody rush hour! Sounds a bit moany I know, but we were pissed off when we were still sat there at 9 a.m. and several people came in for appointments at 9.15 a.m to 10 a.m and they lived 10 minutes walk away! Why didn't they get the bloody 8am appointment?! 
As a number of people have said, it IS OK to complain, because the NHS is NOT free; we pay for it, and the appointments system is a shambles. They act like everyone has got nothing else to do in their life, except sit there waiting for appointments, and will not be inconvenienced or bothered if the appointment is changed again and again! The system is not fit for purpose.