Depending on how you define "account", that can be quite reasonable. In a B2B application, each business customer could get their own SQLite database, and the number of SQLite connections would likely be quite manageable, even though some customers have many users.
Without knowing details about the app, it's hard to know if that would matter. If a small number of concurrent users would ever be using it, I would think it would be NBD.
So now you need one database connection per user...