Eu acho que sua amostra de saída desejada "dividida por tempo" está errada e deveria ser essa
1=> 2014-01-01 23:43:00 - 2014-01-02 02:30:00, as date 2014-01-01
2=> 2014-01-02 02:30:01 - 2014-01-03 02:30:00, as date 2014-01-02
3=> 2014-01-03 02:30:01 - 2014-01-03 03:33:00, as date 2014-01-03
Se for esse o caso, então faça isso
select day, count(*)
from (
select generate_series(
(start_time - interval '2 hours 30 minutes')::date,
stop_time,
interval '1 day'
)::date as day
from t
) s
group by day
order by day
