I have to add my "plus one" to positive experiences. Recently called BA for the first time ever to do a booking over the phone as the website is broken beyond repair. Only a few seconds of waiting (as Silver) and I got through to an excellent agent who was very efficient in making the booking.

The only wrinkle is that I was charged EUR 15 booking fee (only discovered it while reviewing the receipt afterwards), which is not really fair given that the website does not even offer the option I wanted to book (DUB-JFK via LCY with a same-day connection). I ranted about it in a survey I got afterwards, but expecting any refund at this point would be too optimistic .