Yup, my DH has experienced all that almost weekly for the past 8 years since he was diagnosed with cancer which has also caused other conditions including several broken bones meaning he's been under the orthopaedic dept on and off for several years, plus problems with his throat which has meant he has a "ongoing" open file with ENT, again for several years.
It's a nightmare trying to firstly even get an appointment (even though they're always put down as "urgent" due to his cancer/chemotherapy), then of course, there's no option to choose, so when one finally comes through the door, he often has to try to phone to change it, and yes, usually either the phone number on the letter is out of order, or always an answerphone that never gets responded to, or completely the wrong person, or if you manage to speak to the right person, the answer is often that they'll just send yet another "random" appointment that likewise will probably need to be changed. It's virtually impossible for them just to agree a new appointment date/time on the phone. Must create so much extra workload, cause more wasted/missed appointments etc.
It's worse for him because every single month he has the same sequence of appointments for his monthly chemo drugs, i.e. first a blood test, then x days later a consultation with the haematologist, then y days later an infusion, and z days later he has to pick up his oral chemo drugs (inc in a package of lots of others such as anti viral, anti bacterial, etc). Haematologist won't "tick the box" to authorise the infusion/drugs without a blood test for obvious reasons. Same pattern every single month now for 8 years. Every single month, they cock it up, i.e. haematologist appt letter before the blood test appointment letter etc., or drug pick up appt before consultant appointment. Or he turns up on the agreed date to pick up the drugs only to find they've not been ordered! Meaning every single month he has to phone to re-arrange the appointments, of course, with 4 different people! It's apparently impossible to just go to reception and make all 4 appointments for the following month, even though all 4 are held within the oncology department!
The mind boggles!