Max dentro de un período de tiempo con duplicados de fecha

Si lo entendí correctamente, desea contar una entrada distinta para un estado específico en su período de tiempo... si es así, debe usar DISTINCT cláusula en su count() cambiando de conteo (*) a conteo (distinto Entry_id)

with c (Status_Id, Entry_Id, Start_Date) AS (
  select Status_Id, Entry_Id, Start_Date from tbl where
  (End_Date BETWEEN '19000101' AND '21000101')
  AND ((Start_Date BETWEEN '19000101' AND '21000101')
  OR End_Date <= '21000101'))
select Status_Id, count(distinct Entry_Id) as cnt from 
 (select Entry_Id, max(start_date) as start_date from c
  group by Entry_Id) d inner join
c on c.Entry_Id = d.Entry_Id
and c.start_date = d.start_date
GROUP BY Status_Id WITH ROLLUP

EDITAR

Siempre que no le importe qué estado se devuelve para una entrada determinada, creo que podría modificar la consulta interna para devolver el primer estado y unirse al estado también

with c (Status_Id, Entry_Id, Start_Date) AS (
  select Status_Id, Entry_Id, Start_Date from tbl where
  (End_Date BETWEEN '19000101' AND '21000101')
  AND ((Start_Date BETWEEN '19000101' AND '21000101')
  OR End_Date <= '21000101'))
select c.Status_Id, count(c.Entry_Id) as cnt from 
 (select Entry_Id, Start_Date, (select top 1 Status_id from c where Entry_Id = CC.Entry_Id and Start_Date = CC.Start_Date) as Status_Id
  from (select Entry_Id, max(start_date) as start_date from c
  group by Entry_Id) as CC) d inner join
c on c.Entry_Id = d.Entry_Id
and c.start_date = d.start_date
and c.status_id = d.status_id
GROUP BY c.Status_Id

Resultado

Status_id Count
 489       2
 492       1
 495       1