Though I was amused one time when I asked a pupil if he thought he had an accent, and he said "no". He's the only pupil I've ever had with that sort of clipped military delivery typical of people in the British armed forces. Which wasn't surprising, as his father was a captain or something in the armed forces.
tl;dr - it's really hard to asses one's own accent.
I suffer from a condition called "other people's accents". I didn't acquire an accent from my parents other than a tendence to copy my dad's flat "a" sounds for Bath (to rhyme with lath or math) etc. I also went to Drama school where I didn't have to do very much to achieve an RP accent compared to many of my peers.
However, if I am talking to someone, I will, over a short period of time, start to acquire aspects of their accent. My family get very annoyed with this when I'm on holiday - unless we are in France where I *sound* pretty much native, which can come in quite handy. My grammar isn't great - which gives the game away.
Fun anecdote:
Coming back from a mini break to Disney last year, we saw an exhausted Alex Brooker pushing his wife - who at the time was in a wheelchair (I think due to an injury). As we got towards the departure gate he was trying to get through a door, so I popped over and held it for him. As he came through one of the staff thanked me in French, so I just replied "ca fait rien Monsieur" (effectively - don't worry about it or no problem). Alex then also tried to thank me in French and was somewhat bemused when I replied - "oh - it's OK I'm English mate - have a good flight!"