Scheduled backup task doesn't always backup all databases despite always saying job successful
I would add TRY/CATCH blocks to handle errors and log them. The DB could be in single user, being restored or whatever.

Without this, errors can abort in a way that no errors are logged (statement, batch, scope, connection etc)

With TRY/CATCH then everything except for compile or connection aborting errors are logged? but I doubt this is the case.

I'd also use sys.databases which replaces sysdatabases and read more flags:

-- declares etc

BEGIN TRY

    DECLARE db_cursor CURSOR FOR  
    SELECT name, state, user_access
    FROM sys.databases 
    WHERE name NOT IN ('tempdb')

    OPEN db_cursor   
    FETCH NEXT FROM db_cursor INTO @name, @state, @user_access

    WHILE @@FETCH_STATUS = 0   
    BEGIN   

        SET @fileName = @path + @name + '.bak'
        IF @state = 0 AND user_access = 0
        BEGIN
            BEGIN TRY
                BACKUP DATABASE @name TO DISK = @fileName  WITH INIT
            END TRY
            BEGIN CATCH
                -- log but do not rethrow so loop continues
            END CATCH
            WAITFOR DELAY '00:00:20'
        END
        ELSE
           --log user and/or state issues

        FETCH NEXT FROM db_cursor INTO @name 
    END   

    CLOSE db_cursor   
    DEALLOCATE db_cursor

END TRY
BEGIN CATCH
  -- some useful stuff here
END CATCH