Brent recently did a post called In Azure SQL DB, what does “The connection is broken and recovery is not possible” mean? and really the main point of the post was this:
All it really means is, “Click Execute again, and you will be fine.”
FYI you can also get the same error when connecting to an on-premises instance and it still means the same thing.
Also along those lines here is another similar error that also means the same thing:
Translated as: “Click Execute again, and you will be fine.”
You can get additional information here.
I do want to give one warning though. When you see these errors trying again will almost always re-connect you. However, unless your connection is told to go directly to the database you need (as would happen with an Azure SQL DB connection) you may end up back at your default database.