Is it legal to download pirated movies or software for personal---that is non-commercial---use?
Not even when you own a CD/DVD version of the movies/software.
Absolutely not in the US either. Under the normal assumed definition of 'pirated', this is clear copyright infringement regardless of your intentions not to re-distribute material. This could violate any range of federal copyright law pursuant to 17 US code § Ch 1-8 and 10-12 as it would almost certainly fail to qualify as 'fair use'.
It isn't legal.
Here's what might be confusing you: copyright's fair use doctrine does look at whether the use is personal vs commercial. But that's just one thing copyright takes into account. There are a number of other factors courts are required to take into account, by law. See 17 U.S.C. § 107. These other things weigh heavily against you: movies and software are at the heart of intended copyrighted protection; you're taking the entire movie/program; not paying for the items screws up the market for the good; and you're not triggering any exception. If you want to see how the judge has to weigh the factors in practice, see these other Law.SE questions here or here.
Regardless of whether it's legal, you might ask whether the government will enforce it. Copyright owners used to be to prefer to sue the internet provider or cable provider, but that ended up being somewhat ineffective, so owners are now suing individuals who pirate movies/software directly. For example, Breaking Glass Pictures filed a bunch of suits against people who pirated Six Degrees of Hell and K-11. See here. Capitol Records went after an individual for pirating music online. See here.
However, set the law aside for a moment and just look at common sense. The term pirated means "stolen." So we can rephrase your question as follows: "Is it legal to download stolen goods?" It is not.