Getting a boolean from a SELECT in SQL Server into a bool in C#?

As noted in the comments you are returning an integer. You need to return a bit, which ASP.NET will understand as a Boolean.

SELECT A.CompletedDate,
  CASE
      WHEN (@AdminTestId IS NULL AND @UserTestId IS NULL) THEN 
        CONVERT(bit, 0)
      WHEN (@AdminTestId = temp.AdminTestId AND @UserTestId = A.UserTestId) THEN
         CONVERT(bit, 1)
      WHEN (@AdminTestId = temp.AdminTestId AND @UserTestId IS NULL) THEN
         CONVERT(bit, 1)
      ELSE
         CONVERT(bit, 0)
  END AS [Current],

Or as pointed out Damien_The_Unbeliever in the comments below, you could wrap the whole CASE statement in a CONVERT.

SELECT A.CompletedDate,
  CONVERT(bit, 
    CASE
      WHEN (@AdminTestId IS NULL AND @UserTestId IS NULL) THEN 0
      WHEN (@AdminTestId = temp.AdminTestId AND @UserTestId = A.UserTestId) THEN 1
      WHEN (@AdminTestId = temp.AdminTestId AND @UserTestId IS NULL) THEN 1
      ELSE 0
    END
  ) AS [Current],

You could also use CAST(1 AS bit) and CAST(0 AS bit) in replace of CONVERT(bit, 1) and CONVERT(bit, 0) respectively (see T-SQL Cast versus Convert for more information).

You could also the conversion client-side:

using (SqlDataReader reader = command.ExecuteReader())
{
    if (reader.Read())
    {
        var dto = new GetTestsDTO();
        dto.Current = Convert.ToBoolean(reader.GetInt32(1));
    }
}

The easiest way (SQL Server side) is to convert values 0 and 1 to BIT datatype:

SELECT  A.CompletedDate,
  CASE
    WHEN (@AdminTestId IS NULL AND @UserTestId IS NULL) 
    THEN CAST(0 AS BIT)
    WHEN (@AdminTestId = temp.AdminTestId AND @UserTestId = A.UserTestId) 
    THEN CAST(1 AS BIT)
    WHEN (@AdminTestId = temp.AdminTestId AND @UserTestId IS NULL) 
    THEN CAST(1 AS BIT)
   ELSE CAST(0 AS BIT)
  END AS [Current], 

or entire expression at once:

SELECT  A.CompletedDate,
    CAST((CASE
         WHEN (@AdminTestId IS NULL AND @UserTestId IS NULL) THEN 0
         WHEN (@AdminTestId = temp.AdminTestId AND @UserTestId = A.UserTestId) THEN 1
         WHEN (@AdminTestId = temp.AdminTestId AND @UserTestId IS NULL) THEN 1
         ELSE 0
         END)
         AS BIT) AS [Current], 

---

SQL Server Data Type Mappings:

╔═════════════════════════════════╦═════════════════════╦═══════════════════════╗
║ SQL Server Database Engine type ║ .NET Framework type ║ SqlDbType enumeration ║
╠═════════════════════════════════╬═════════════════════╬═══════════════════════╣
║ bit                             ║ Boolean             ║ Bit                   ║
╚═════════════════════════════════╩═════════════════════╩═══════════════════════╝

As the previous comments suggest, SQL cannot return the "boolean" value, and you are not returning it either.

My suggestion/solution (that I use, and have seen used everywhere I worked) is to simply recover that Integer in your object, and then use a 'map' function that will transform that Integer to a Boolean.

The (simple) method will do something like this (attention; Java version ahead):

public static boolean integerToBoolean(Integer myInt){
     return myInt == 1 ? true : false;
}

Best of luck.