Yes, it happens, although it is less common than hiring people in the standard undergraduate, MBA, or early experienced-hire windows. At 39, firms are likely to look much more closely at how your prior experience translates and what level makes sense.

The important issue is not whether 39 is an acceptable number. It is whether your experience creates a credible reason for the firm to hire you now. Strong leadership, industry depth, commercial exposure, and evidence that you can solve unfamiliar problems all help.

You should also go into the process with realistic expectations about title and progression. An experienced hire can enter at a level that does not perfectly match the seniority of their current role, because consulting firms evaluate experience through their own career ladder.

If the opportunity still makes sense after you understand the likely level, I would apply rather than trying to predict the outcome from age alone.

Yes. I know one person who joined at 38 and another around 40. Both were experienced hires with pretty strong backgrounds.

Imo the bigger question is what level they would slot you into.