Thank you for your question.
Yes they can, unfortunately, if they can show that the employee would have been discharged anyway in the absence of the leave request. To succeed on an interference claim (that is, a claim that the employer interfered with the employee's rights to take FMLA time by terminating him/her), an employee must show: (1) he or she was entitled to FMLA leave; (2) an adverse action by the employer interfered with the employee's right to take FMLA leave; and (3) the adverse action was related to the exercise, or attempted exercise, of the employee's FMLA rights.
If there is sufficient evidence to suggest that the employer's motives for termination
were genuine -e.g., the employee had numerous bad reviews, had been absent from work or tardy regularly, had been insubordinate, etc. - and therefore, not related to the FMLA request it would not be considered a wrongful termination
If there is no reason to suggest that an employee would not have otherwise been terminated then it is possible that an employee may have a claim for wrongful termination/retailiation, and that is something that would want to discuss more in detail with an employment lawyer. For the employer, it all comes down to proper documentation. Legally, nothing prevents them for terminating an employee after they request FMLA (or even while an employee is on FMLA, for that matter) - but the timing is rather poor -thus documentation as to why they took the action they did is imperative.