A B737NG Flight Crew reported problems with their flap load relief system on approach; leading to a deviation to a nearby airport with a longer runway.

Narrative

On approach flaps would not extend beyond 25 degrees. Autopilot and autothrottles were engaged and using VNAV procedures. Passing FAF checked speed below 200 KTS and called for gear down and flaps 15. Checked speed below 190 and called for flaps 25. Checked speed below 175 and called for flaps 30 followed by flaps 40 when speed below 162. I noticed it took an unusually long time for the speed to bleed below 162. Then noted that the auto speed bug was still indicating flaps 25 speed and noted the flaps were still indicating 25 degrees. Reselected flap handle to flaps 25 position momentarily and then back to flaps 40 position. Flaps remained indicating 25 degrees. Executed a missed approach and due to our low fuel state elected an immediate divert to [a nearby airport] for flaps 15 landing. Landed normally using flaps 15 landing procedures. This aircraft had two previous flap problem write-ups and flap maintenance had been performed on this aircraft prior to the previous leg. Aircraft was signed off after this event as could not duplicate (on ground). Upon review of the flap load relief system; I believe the system on this airplane may be malfunctioning causing a flap lockout at higher than intended speeds. Check the load relief system on this aircraft.