The answer to this question depends greatly on the temperature of the hot bath, the ambient temperature of the bath room and whether it is well ventilated, and the tolerance of the person in the bath, including his body characteristics such as body mass index, and the surface area of his body.

The most probable answer is no. To initiate hyperthermia on a moderate temperature usually involves more than 10 minutes. This is sufficient for water to be cooled by 1 or 2 degrees. Sugnificant discomfort will have encouraged the person to come out of the bath, quickly lowering the body temperature well before the person passes out.  While cannot be directly compared, a sauna session can reach up to 80 degrees celcius, aided by perspiration, most healthy adult can endure up to 15 minutes.

Personally I have been into 40 degrees Celcius hot spring in Japan for 15 minutes, it gave a similar experience as being a bit drunk and dizziness ensues, but I did not appear to be very ill.