Yes. Just download it from the Adobe site and use the serial number that came with your discs.
I tried but before I can download it I have to buy it online... should I download the test version? It is possible only separately... elements and premiere but I have ONE KEY with the DISC for both ...
I was in the same boat as you, so I went to Best Buy and purchased the box (on sale, by the way). I then downloaded the trials separately for Photoshop Elements and Premiere Elements from the Adobe product site.
In the box you should have been given 2 discs (one for Photoshop Elements (PS) and one for Premiere Elements (PE)) and something called a "redemption code" on a square of colorful cardboard with instructions on how to get your serial keys for installation. The single redemption code is NOT the serial/product key you use to install your purchase. Once you follow the instructions on this card you will be shown two serial/product codes, one each for PE and PS. Write them down, print them, whatever, you will need them during installation.
Now go to the downloaded trial file(s) and run the installer, either for PE first or PS, it makes no difference. During the installation you will be asked if you want to start a trial or use a serial/product code. Choose "I have a code". Enter the product key and you will be golden. Repeat for the other program.
Hope this helps!