At this point, it would be extremely difficult. Nintendo has a following, they have established properties and franchises, and they have a huge amount of resources invested in keeping it going.
I'm not sure they've ever had any real contenders, though. First, it would take a massive amount of money and, second, dedication that matched that of Nintendo. Sony is the only one to match them when it came to what they could invest in the market but they didn't come close in terms of dedication. Handhelds have always been second-fiddle to Sony. They've never tried to generate the same amount of handheld content as Nintendo and thus were not serious challengers. I don't think they ever assumed they would "win" that market, they just wanted a share of it.
Only an enormously wealthy company with capital to invest in hardware, marketing, and studio development would have a shot. Oh, and they would have to be Japanese.
