The actual, exact text of the error is
The query did not run, or the database table could not be opened.
The sheet in question was running a stored procedure, which had been working fine, albeit very slowly. A coworker tuned the proc and happened to use a temp table to accumulate the results for multiple queries instead of a UNION, producing better execution plans for each. However, despite being 4x faster, the proc stopped running from Excel, throwing the error message above.
At first I thought there might be a problem with the final DROP #table statement in the proc, so I removed that. No help. I adjusted various properties of the database connection from Excel. No change. I fiddled with the table setup in Excel. Nothing.
Then I remembered that I had some difficulties with pyodbc a while back that were resolved by setting NOCOUNT ON in the query. I can't recall the exact cause, but it's something about the driver seeing the first count returned as the query result, when of course it's not in a proc that populates a table with a few queries and then selects from it as the final step. Anyway, I set NOCOUNT ON at the beginning of the proc and OFF at the end and voila, it worked. So try that if you're having difficulties with Excel not executing your stored proc correctly.