You can do it, but you need an associated app that can handle PDFs. I've been able to open the PDF in Dropbox, then save it to my Dropbox account. I have PDFPen on my devices, and I think it is possible to open and save in that app too, but I have not tried it. Bottom line is that you need an app to handle the PDFs before you can save to it.
Usually to do something like this you need an app that's associated with the PDF to allow you to open it once it's downloaded. I have an app called ReaddleDocs (I still have the iPhone version, not sure if there's na iPad version).
It has a browser that I can use to download, and also opens PDF files.
There are other apps out there that open pdf files, GoodReader and an Adobe reader among them. No idea if they handle downloads.
I was assuming you were referring to downloading a PDF from a website. The procedure I outlined above using "open in" Dropbox will allow you to save that PDF to Dropbox. From there, you can switch to the Dropbox app on your iPad and open and read the document from there. Taking it a step further, PDFpen will allow you to annotate or edit the PDF document.
All that said, I do keep a good number of PDFs in my dropbox account. My dropbox account is linked to my iPad and my iPhone, so I have quick access to these whichever device I choose to use. For example, I have a number of recipes in PDF format in my dropbox account. I can easily access these from my iPad when I am in the kitchen preparing dinner. I can also access them from my phone, for example when I am at the market shopping for food and I forget the particular ingredients in a recipe. I can simply open dropbox, look at the recipe and purchase the correct ingredients. It all works very very well.