The Rockingham Tea Set referred to in the title is a set owned by some new employers of practical nurse Grace Kelly who was cast in one of the many television dramas she did before becoming a film star ever so briefly.
As the tea is served Grace relates in flashback the story of the bad time she had as a nurse for paraplegic Louise Allbritton. Allbritton's role was that of a bitter woman who was paralyzed and taking it out on the immediate world which includes her loved ones and any hired help like Grace that's around.
The duel between Kelly and Allbritton is what's featured in this television drama. I can't reveal too much about how it ends, but here's a big old hint, think about the plot for the Otto Preminger classic, The Man With The Golden Arm.
This television drama was directed by Franklin J. Schaffner who would go on to win an Oscar for Patton. I can't help thinking that the petty cash account for the Patton production probably is what the whole budget of The Rockingham Tea Set was. But live television drama was a great training ground for those who worked behind the camera as well.