Three Cuppes Lane is in the centre of Salisbury.
It was named fairly recently after the Three Cuppes Inn, which is now a private house in Winchester Street. The Three Cuppes themselves could have a couple of different meanings:
- Three Cups’ are the symbols of the Salter’s Guild 1. It may be relevant here that the opposite side of the chequer2 from where the Three Cuppes Inn stood is Salt Lane. I think that there are Three Cups visible in the current badge of the London ‘Salters Company’ on the website
- The ‘Three of Cups’ is a tarot card, supposedly representing something like completion, or conclusion, or fullfillment
As noted elsewhere, in pre-literate times it was important for Inns to have easily recognizable signs above the door. The Three Cups would be a comparatively easy pub sign to paint or make – and fits in with the purpose of a pub.
There are still ‘Three Cups’ pubs in Dover 3 and Stockbridge 4
Footnotes
- http://en.wikipedia.org/wiki/Pub_names [↩]
- A ‘chequer’ is a ‘block’ – Salisbury is laid out in a grid or ‘chequer-board’ pattern [↩]
- http://www.thethreecups.co.uk/ [↩]
- http://www.the3cups.co.uk/ [↩]

You’re correct in saying that the Three Cups is connected with the Salters Company, which still exists today and sponsors education in industrial chemistry.
I was also a favourite heraldic device representing the office of the King’s Butler, an official appointment rather than of a servant.
Elaine Saunders
Author – A Book About Pub Names
http://www.completetext.com
[...] Three Cuppes Lane, on the same chequer (city block), and the Three Cuppes Inn are named in reference to The Salters Company, whose emblem includes the three cups. [...]