Sorry but can you clarify that somewhat. When yous ay borders do you mean the borders applied using Format / cells / Border? If so then if you copy those cells you will automatically get the borders copied with them and need to lose them by using a paste special method.
Now that having been said, if you are talking about for example a border around a block of cells, and not borders inside the block then this is possible. Almost any border you see on a sheet has two sides to it, and you can apply a border to either side, eg
The border between cells A2 and A3. This can EITHER be a bottom border applied to cell A2, OR a top border applied to cell A3, OR you could apply it to both.
So to answer your question, assuming you had the range B10:D20, if you actually selected that range and applied a border, then copying that range will take the border with it. If however, you apply a border to
The bottom of B9:D9
The right of A10:A20
The top of B21:D21
The left of E10:E20
You will see the same as before, BUT, you can now copy the range B10:D20 and you will not be copying the borders as well.
Like I said though, if you have borders in between then you're out of luck.