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Desired range:              +---------------+
Case 1,2:          +-1-+                        +---2--+
Case 3,4:             +----3-----+      +--------4-----------+
Case 5:                        +---5----+
Case 6:            +------------------6----------------+

select * from table 
where (start_time    between desired_start and desired_stop)
or    (end_time      between desired_start and desired_end)
or    (desired_start between start_time    and end_time)
or    (desired_stop  between start_time    and end_time)