T, Sql Server - rextester

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declare @t table (Name varchar(100),  Datetime_Start  datetime,  Datetime_End datetime);
insert into @t values
 ('A'   , '2017-01-02 00:00' , '2017-03-28 00:10'),
 ('A'   , '2017-05-14 23:50' , '2017-05-29 23:50'),
 ('B'   , '2017-05-18 00:00' , '2017-05-18 04:00'),
 ('B'   , '2017-05-18 02:00' , '2017-05-18 03:00'),
 ('C'   , '2017-01-02 00:00' , '2017-01-17 15:50'),
 ('C'   , '2017-01-14 03:50' , '2017-01-28 15:50');

with Datetime_Starts as 
( 
  select distinct name, Datetime_Start 
  from @t as t1 
  where not exists 
    (select * from @t as t2 
     where t2.name = t1.name 
       and t2.Datetime_Start < t1.Datetime_Start 
       and t2.Datetime_End >= t1.Datetime_Start) 
), 
Datetime_Ends as 
( 
  select distinct name, Datetime_End 
  from @t as t1 
  where not exists 
    (select * from @t as t2 
     where t2.name = t1.name 
       and t2.Datetime_End > t1.Datetime_End 
       and t2.Datetime_Start <= t1.Datetime_End) 
) 
select name, 
       Datetime_Start, 
      (select min(Datetime_End) 
       from Datetime_Ends as e 
       where e.name = s.name 
         and Datetime_End >= Datetime_Start) as Datetime_End 
from Datetime_Starts as s;

View schema
Execution time: 0,02 sec, rows selected: 4, rows affected: 6, absolute service time: 0,17 sec