Many individuals within BA really care about customers, but collectively, the organisation often fails to deliver on customer service and support. Running a big airline is about a lot more than just flying planes.
There was a time when BA was my default choice of airline. Now it's just one of the crowd. That's not down to the service on board - although some Heathrow cabin crews have been particularly sour-faced lately. It's down to the whole end-to-end experience, including how the company deals with "unexpected" problems.
I left BA (EC Gold at the time) due to their call centre's poor handling of a reroute the day before one of their strikes. I won't boar you with the details but this, and the crappy (non) follow-up by their (non) customer service, means I haven't flown them with a revenue ticket since. (I did burn through my miles though)
And from the comments on this thread, I see no reason for going back either.