May an employer classify an employee as nonexempt, even if the employee's duties qualify for an exemption?

Yes. There is no requirement to ever make any position exempt. Employers need not automatically designate as exempt employees who clearly perform the primary duties of the job functions described under one of the exemptions. 

Employers who maintain strict control over requests for overtime work, for example, or who need to keep track of all employee hours for billing reasons may find that payment on an hourly basis works best for them. In such cases, the workers will most likely be nonexempt by default, simply because they are paid on an hourly basis, as many of the Fair Labor Standards Act (FLSA) exemptions require an employee to be paid on a salary basis to qualify as exempt.

Another way to think of this issue is to remember that employees typically are presumed nonexempt until proven to be exempt, meaning that all employees are covered by the FLSA’s minimum wage and overtime requirements until an employer chooses to prove they meet the special criteria for one of the exemptions.

The advantage of using the exemptions, of course, is the ability to avoid the cost of overtime pay for additional hours of work. However, if an employer cannot clearly show that a particular employee’s duties and pay qualify for exempt status, the employer should classify that person as nonexempt.