I see three quite different types of behaviour all rolled up into this one overused word.
First, there is the poor performer who has skilfully squirmed out of performance reviews for many years with a succession of managers too timid to confront. A new boss joins, looks askance at the miserably inadequate achievements of this person and steadily applies due process: tough but fair feedback.
This has no immediate effect - why should it? The person has learnt that there are no consequences to doing 'near enough is good enough' work. When, to their horrified amazement, it is clear that their job is in danger, this person first goes off with that wonderfully vague condition, 'stress', and then squeals, 'Bully!' Don't waste your energy worrying about this accusation: it is easily dismissed with hard evidence. (Please do not e-mail me to tell me that you should involve your HR person. Of course you should.)
More concerning is the situation where there are differences of opinion and the more junior person accuses the more senior of bullying. It is bullying if you are constantly mistaking being coercive for being authoritative, arguing for discussing, sarcasm for humour and over-tough pressure for firmness.
You are at risk of bullying if your range of influencing tactics is limited. Bullies interrupt rudely, because they want theirs to be the opinion that counts and do not know how to make sure that this is what happens. It is bullying if you dismiss or just do not notice the importance of people's feelings. It is bullying if your comments seem personally hurtful and if your staff have an unhealthy obsession with dodging your moods. I worked with one client who discovered his team ran a daily mood forecast with a book on how often the weather would be stormy.
It is a bad sign if, when finally called to account, you turn yourself into a victim - your accuser becomes the persecutor and it is their fault for triggering your behaviour. This is denying responsibility for your effect on others: the adult equivalent of saying, 'a big boy did it and ran away'. If this is you, it is potentially a career derailer. Get proper 360-degree feedback, then a relentlessly tough but forgiving coach and face up to the need to change your behaviour.
Finally, there is the born bully, thankfully a rare creature. This is someone who can present themselves with charm much of the time, but who has a constant need to prove their own superiority by finding an apparently 'weaker' person to belittle. This is someone who takes a cool delight in well-hidden processes of hurting and tormenting others. My experience is that there is little hope of change for this person. But then he or she will not be reading this column.