When I install the game I receive the error 'Cannot connect to download server. Please check your internet connection'.
This indicates that the BioShock installation has encountered a problem connecting to the internet when attempting to contact the download server.
As stated in the minimum requirements a Internet connection is required so ensure that your modem/router is connected and working.
UPDATE 6/1/2012: We are aware of an issue causing this error, despite having an active internet connection. We are working on resolving it.