I've got two data points for you:

1) About 3 years ago returning from CPH - got to the airport well in time for the earlier flight, just hand luggage. Point blank refusal at checkin to let me on the earlier flight (non flexible ticket) and then again at the gate. Fair enough really - but I was then airside and there was tons of space on the flight (confirmed by a work colleague). Not sure it would have cost BA to let me on.

2) Just recently - returning from TXL - non flexible again. 4 colleagues on an earlier flight. Flightstats suggesting that economy is wide open. Asked at the gate this time - they said no initially - ticket sales might be able to help, but ticket sales are back through security. So suggested I go back land side. Spoke to colleagues and said I'd go back through, ask, and if not, go to the lounge (also land side for anyone not familiar with TXL!) as my flight was delayed. As I went to ask the gate girls how to get out, they told me to hold on, commented I was silver, and moved me. They then claimed it was due to the delay on the later flight having 'released' availability to move me.

One other (non status) colleague got the same treatment. Appeared to be some form of 'emergency' policy.