You most definitely can, but you should remember that filing for a patent is very expensive. :)
This was years ago, so I cannot remember all the details, but I did attend one or two patent workshops/talks, so I think can can weigh in on this.
Basically, there are a few criteria for a certain submission to be patentable, the most important of which was novelty. There were some others, such as not simply combining exiting technologies to create something obvious (IIRC something called 'inventive step' or something like that), but largely it revolved around it being new.
As you might imagine, the above is somewhat interpretable, so a lot of these could go through if they were a tiny bit novel in their approach. Even if your idea was not patented, your idea might still be published as a white paper just so that others could not then invent something similar. By exposing that idea to the outside world, it became public knowledge, so HP could not go and patent something similar.
I remember being told it's all about the mindset. You don't have to be particularly bright or have some 'divine' spark, you just needed to take a problem/functionality/thing and turn it fiddle with it until you get something new, then do some research (there was special software for this) to see if it already exists, get someone internal to review it (a Master Inventor, as they were called), get in touch with the internal lawyers and send it for review.