Here's one way that finds first month and last month (when the last month has 'Bad' ) for each name, then joins that to a list of all months from the earliest date in the data until today, then uses OUTER APPLY to find the previous value to determine what to fill the gaps with.

Working example in this db<>fiddle - including the edge case where a person's latest Filter = 'Good' .

DECLARE @BeginOfMonth date = DATEADD(DAY, 1, EOMONTH(GETDATE(),-1));

;WITH Origin AS 
( /* let's get the data and fix the year/month column */
  SELECT Name, Filter, Data, 
    [Month]   = DATEFROMPARTS(YearNumber, MonthNumber, 1),
    LastMonth = MAX(DATEFROMPARTS(YearNumber, MonthNumber, 1)) 
                OVER (PARTITION BY Name)
  FROM dbo.whoknows /* we don't know your table name */
), Agg AS
( /* let's get the first and last month for each name */
  SELECT Name, S = MIN([Month]), E = COALESCE
         (MAX(CASE WHEN [Month] = LastMonth AND Filter = 'Bad' 
         THEN [Month] END), @BeginOfMonth)
  FROM Origin GROUP BY Name
), AllMonths(m) AS
(
  /* let's get ALL the possible months to fill gaps */
  SELECT MIN(s) FROM Agg UNION ALL
  SELECT DATEADD(MONTH, 1, m) FROM AllMonths
  WHERE m < @BeginOfMonth
)
SELECT Agg.Name, 
  [Month] = DATENAME(MONTH, am.m), 
  [Year]  = YEAR(am.m), 
  Filter  = COALESCE(o.Filter, GapFill.Filter),
  Data    = COALESCE(o.Data,   GapFill.Data)
FROM AllMonths AS am 
INNER JOIN Agg ON am.m >= Agg.S AND am.m <= Agg.E
LEFT OUTER JOIN Origin AS o
  ON o.Name = Agg.Name AND o.[Month] = am.m
  OUTER APPLY 
  (
    SELECT TOP (1) org.Filter, org.Data
    FROM Origin AS org
    WHERE org.Name = Agg.Name AND org.[Month] < am.m
    ORDER BY org.[Month] DESC
  ) AS GapFill
ORDER BY Agg.Name, am.m OPTION (MAXRECURSION 1000);

Output: