Have you verified that there is in fact a row where Staff_Id = @PersonID? What you've posted works fine in a test script, assuming the row exists. If you comment out the insert statement, then the error is raised.
set nocount on
create table Timesheet_Hours (Staff_Id int, BookedHours int, Posted_Flag bit)
insert into Timesheet_Hours (Staff_Id, BookedHours, Posted_Flag) values (1, 5.5, 0)
declare @PersonID int
set @PersonID = 1
IF EXISTS
(
SELECT 1
FROM Timesheet_Hours
WHERE Posted_Flag = 1
AND Staff_Id = @PersonID
)
BEGIN
RAISERROR('Timesheets have already been posted!', 16, 1)
ROLLBACK TRAN
END
ELSE
IF NOT EXISTS
(
SELECT 1
FROM Timesheet_Hours
WHERE Staff_Id = @PersonID
)
BEGIN
RAISERROR('Default list has not been loaded!', 16, 1)
ROLLBACK TRAN
END
ELSE
print 'No problems here'
drop table Timesheet_Hours