Recently I made a journey that involved me going from Paddington to Victoria to catch another train from there. However, I wanted to break my journey within London for the day, which I obviously could easily have done at either Paddington or Victoria. However, I wanted to make the break at Kensington, along the underground part of the journey, and found that my ticket wouldn't open the barriers there, and was told by the station staff at the barrier that a break in journey wasn't allowed with a rail ticket there.
Can someone let me know if that is the case?
Yes, that is correct. Cross London (Maltese cross) tickets are valid for a single unbroken journey between a defined list of a London interchange stations. (counting a return as 2 tickets, 2 journeys). South Ken, West Ken and High Street Ken are unlikely to be in the list, not sure about Ken Olympia.