I recently had this problem - it was incredibly frustrating not to be able to understand what was going on. I still don't know what caused it, all I know is what resolved it and am posting here in case it helps someone else.
The model data source which was causing the problems used the '.NET Framework Provider for SQL Server'. I tried a million things and was always returned the IDbConnection error as above.
In the end I created a new and simple test tabular model from scratch which connected to the data source just fine. When I checked the data source details, it had defaulted to using provider 'SQL Server Native Client 11.0'. So I changed the provider on the problematic model, deployed it and whaddya know... it connected successfully.
As I say I don't know the whys & wherefores but my guess is an update did something to .NET which caused the problems?