Following this Boeing 720 (a 707 variant) accident in 1963 the investigators determined that the stabilizer trim stalled for exactly that reason. In that case it was the crew who ran the trim well away from the neutral column force position in response to a strong updraft, at least according to what the investigators theorized from the limited data recorder info. My understanding is that the 737 stab control system design was based on the 707 system, but I don't know the system force differences between the models at extreme out of trim conditions.
https://lessonslearned.faa.gov/ll_ma...abID=1&LLID=66