Assuming there are NO blank rows in the data this should deal with
most of your problem:
Sheets("Sheet1").Range("A16").CurrentRegion.Copy Sheets("Sheet4"). _
Range("A65536").End(xlUp).Offset(1, 0)
Sheets("Sheet2").Range("A16").CurrentRegion.Copy Sheets("Sheet4"). _
Range("A65536").End(xlUp).Offset(1, 0)
Sheets("Sheet3").Range("A16").CurrentRegion.Copy Sheets("Sheet4"). _
Range("A65536").End(xlUp).Offset(1, 0)