program Copia;
\{\$APPTYPE CONSOLE}
uses
SysUtils;

procedure FileCopy(const FromFile, ToFile: string);
var
FromF, ToF: file;
NumRead, NumWritten: integer;
Buf: array[1..2048] of Char;
begin
        AssignFile(FromF, FromFile);
        Reset(FromF, 1);		{ Record size = 1 }
        AssignFile(ToF, ToFile);	{ Open output file }
        Rewrite(ToF, 1);		{ Record size = 1 }
        repeat
                BlockRead(FromF, Buf, SizeOf(Buf), NumRead);
                BlockWrite(ToF, Buf, NumRead, NumWritten);
        until (NumRead = 0) or (NumWritten < NumRead);
        CloseFile(FromF);
        CloseFile(ToF);
end;

procedure FileCopyAdd(const FromFile, ToFile: string);
var
FromF, ToF: file;
NumRead, NumWritten: integer;
Buf: array[1..2048] of Char;
begin
        AssignFile(FromF, FromFile);
        Reset(FromF, 1);		{ Record size = 1 }
        AssignFile(ToF, ToFile);	{ Open output file }
        If fileexists(Tofile)
        then
        Begin
                Reset(ToF, 1);
                seek(ToF,filesize(ToF));
        end
        else Rewrite(ToF,1);
        repeat
                BlockRead(FromF, Buf, SizeOf(Buf), NumRead);
                BlockWrite(ToF, Buf, NumRead, NumWritten);
        until (NumRead = 0) or (NumWritten < NumRead);
        CloseFile(FromF);
        CloseFile(ToF);
end;

begin
        FileCopy('prueba1.dat','final.dat');
        FileCopyAdd('prueba2.dat','final.dat');
end.

El programa es para consola, simplemente es de ejemplo, para que veais como va.