I was able to recreate the problem periodically enough (but not always) after two rings on incoming calls even with one particular phone directly connected to the Obi200 using a short phone cable (all of the house wiring bypassed). The phone + Obi200 combination exhibiting this problem is a General Electric Big Button CID Speakerphone.
https://www.amazon.com/29369GE1-Button-Desktop-Waiting-Speakerphone/dp/B00079VIFUWhile I was hearing ringing on the calling cell phone, on the GE phone's receiver I heard silence plus a sort of clicking/scratching sound every few seconds (probably corresponding with the continued ringing despite being off-hook) for a few iterations during which the Obi200 web interface, showed a Status -> PHONE Port Status of Ringing (with LoopCurrent 19 or 20 mA, VBAT 14 V or 15 V, TipRIngVoltage 5 V or 6V) followed by a State of "On Hook" (while I was still off hook) with the same other parameters (LoopCurrent 19 or 20 mA, VBAT 14 V or 15 V, TipRingVoltage 5 V or 6V). Again, if we let this phone ring for three or more rings, going off-hook consistently results in the call being answered properly.
Another phone for which the problem consistently does not occur when answering shows, when off-hook, LoopCurrent either 19 or 20 mA, VBAT either 13 V or 14 V and TipRingVoltage 4V (while attached to the house wiring, in this case close to the Obi200).
While the phones are all On Hook, the Port Status is On Hook, LoopCurrent 0 mA, VBAT 55 V, TipRingVoltage 46V.
The Obi200 is hardware version 1.4, software version 3.2.2 (Build: 5898EX). There are no optional USB dongles attached. It is Ethernet wired to the router.
The port settings are all default (600 ohm, OnHookTipRingVoltage 46, OffHookCurrentMax 20, etc).
I will be back here at Christmas. What can I do to troubleshoot this problem further or might I have provided enough information now to allow someone to tell me what may be wrong and ideally what to configure to correct it?