To answer the question in your title, yes you can. It's done all the time by some morally bankrupt people who start companies, run-up debts and then liquidate owning their suppliers thousands before starting again with a subtly different name, maybe even buying the previous companies assets for a nominal sum (but not the debt!)
Nothing to stop you being a Director of another company in this way (or, indeed, several companies at the same time) unless you are legally disbarred from serving as a Director by the authorities.