Use Distinct With Sum?
I think what you're looking for is an aggregate on employeeID where the CASE goes inside the SUM() aggregate function.

SELECT employeeid,
       SUM((CASE WHEN [event] = 'Walk' THEN [time] ELSE 0 END)) AS walktime,
       SUM((CASE WHEN [event] = 'RUN' THEN [time] ELSE 0 END)) AS runtime,
       SUM((CASE WHEN [event] = 'Sleep' THEN [time] ELSE 0 END)) AS sleeptime
FROM #Bobsled
GROUP BY employeeID;


This way, you get one result row per employeeID, basically pivoted on event.