I added a measure based on a fact table column that had never been used in the cube before. ... [T]his column wasn't included in the index of the table, so SSAS was having to perform a full table scan when processing, whereas it had previously used the index.
So after rebuilding the index on the DWH table to include that column ...
So one of the two was an aggregation. That would be one example I could understand. Not really following the situation you're describing. For example, unsure what you mean by "the index of the table." Would you be able to provide more details?