How do you measure the number of open database connections

You can try running a query against the master db like this:

SELECT SPID,
       STATUS,
       PROGRAM_NAME,
       LOGINAME=RTRIM(LOGINAME),
       HOSTNAME,
       CMD
FROM  MASTER.DBO.SYSPROCESSES
WHERE DB_NAME(DBID) = 'TEST' AND DBID != 0 

See this link for more details.

Have you tried running the sp_who stored proc? If there are stale open connections they should show up there.

To show just the sa users processes run:

EXEC sp_who 'sa'