Yes, you can be denied time off legally. If you have no PTO available to you, then your employer can prohibit you from taking time off. The fact is that most employment arrangements are what is know as "at will". This means that a company can set the conditions of the workplace much as it sees fit. That is unless a worker's treatment constitutes some form of legally actionable discrimination or violates the terms of an employment contract or union agreement. While unfortunate, your choices are to either accept the situation, complain and risk termination, or quit.