If you're indeed using 64bit Background or 64bit Server you may have some extra work to do.
If you're straight up using ArcMap 10.1 on a 64bit machine, this should "just work". ArcMap and Python are 32bit, thus if it ArcMap connects, Python (32bit) will as well.
If you're using a 64bit product (BG or Server) you'll need to ensure you have the 64bit client libraries to connect.
I believe if you're connecting to SQL Server, both the 32 and 64 bit get installed together (so either works). If you're using Oracle, you'll need both 32 and 64 bit versions of the instaclient and properly set up the information in the PATH variable to 64 first, 32 second (assuming you're using the instaclient). Those are the database flavors I know off the top of my head, instructions probably vary depending on other types.
edit: whats the full Python command you're using in your script? Are you doing:
arcpy.MakeFeatureLayer("c:\data\mysde.sde\sde.FC.pts" .... ) or
arcpy.MakeFeatureLayer("database connections\mysde.sde\sde.FC.pts" .... ) ?