Employees are either entitled to overtime — non-exempt — or exempt from it.
What decides it
Three things together, and all must hold for exempt status: the person is paid a salary, that salary is above a threshold, and their actual duties fall within a recognised exemption.
Job title is irrelevant. So is calling someone a manager. What counts is what they do most of the time.
Where trade businesses get it wrong
A working supervisor who spends most of their week doing the same work as the crew is generally non-exempt, however they are paid and whatever the title says.
Paying a salary does not by itself make someone exempt. A salaried non-exempt employee is still owed overtime for hours beyond the threshold.
Why it is expensive to get wrong
Liability is for unpaid overtime going back years, frequently doubled, plus legal costs. It is assessed on records the employer is required to keep, and where those are missing the employee's account generally stands — see record retention.
The separate contractor question
Whether someone is an employee at all is a different test again, and misclassifying an employee as a contractor carries its own liability for unpaid taxes and benefits — see employee or contractor.
Both tests are applied to the facts of the arrangement rather than to the paperwork.
The thresholds move
The salary threshold is set by regulation and has changed several times, and several states set their own, higher, figures which take precedence.
Because of that this article names no number. The current threshold, and whether a given role qualifies, is a question for an employment attorney or a payroll provider in the jurisdiction — and it is a cheap question relative to the exposure.