Charles:
Ken didn't say he had a router table, but he did mention "dropping the wood down on the bit in a router table..." By him saying it like that, I would agree that he doesn't have the experience level to do something like that. He said he had a plunge router... He doesn't have his profile filled out, so we cannot guess beyond that.
The title is a little misleading, as instead of creating an actual "stopped groove" or "blind dado," he just needs the groove to not continue to the outside edges of the assembled frame. Otis caught that...
Ken:
Along with Otis, it would be good to know the tools you have, your experience level and what you are building as the finished product. From the way you describe it, you say you wanted to create a frame with a groove in just the top piece and bottom pieces.
The confusion is in your familiarity of nomenclature, so how about if I try to tranlsate and tell me if I'm right or wrong, to get a "common understanding." That is what communication is, an understanding between both sides of the conversation. Right?
Here goes-- You want to build a frame for something... (What purpose would fill in some blanks?) It will have 4 sides. It will have 2 grooves, one in the top and one in the bottom... (Are the grooves on the back of these pieces or on the insides of these pieces?) These grooves will be 1/4" deep... (how wide and what will go into the groove, their purpose?)
Biggest mystery to me in your post... Is what do you mean by "to build up the thickness of the wood?"
The reason it is not a stopped groove in reality, like Otis picked up on... is with a mitered frame any inside groove can be machined in it's entirety without it extending to the outside. If it is a rail and stile construction, grooves tooled full length into the rails would not extend to the outsides (would stop where it meets the stiles.) A stopped groove is where you start a groove away for the edge of a workpiece and end the groove before the other edge, where the groove is topped before the material ends. In how you described, the tooling can go over and edge, just not when it is assembled into the finished piece.
Sometimes, a sketch or picture can be a visual aid to bring across an idea and give clarity.