What was your previous version of Excel?

Are you using CTRL+C to copy? If so, try again using right mouse click copy and make sure you see the dancing border on the selection you're copying. Is it possible you think you're clicking CTRL+C and it's not "taking"? Just to make sure that's not it.

Are both your source files and destination files saved in the current .xlsx format?

When you are copying a range of cells from workbook 1, are you selecting just one cell (the upper-left-most) in your destination workbook, or another range? I think you must select just one cell and it pastes in the appropriate number of adjacent ones.

Can you check that your destination workbook/worksheet are not protected in any way?

Are you pasting where an area is blank and there are no merged cells? Or over the top of existing data?