Other than intercontinental travel, where *A rules would require it (and even then it's occasionally like puling teeth), there are no hard-and-fast guidelines. However, your chances are much better if the delay is UA's fault -- mechanical or crew scheduling rather than weather or ATC.

Also, for misconnecting at the origin, it would probably need to be a pretty severe delay with the connecting flight showing as on-time. SOP in that situation is to send you to the connecting point and tell you that your connection is probably delayed too -- which, to be fair, is frequently true.