Going back to the start of last gen, Sony screwed up. PS3 launch was horrible and as a result it made room for the 360 to become a huge success. Microsoft deserves a lot of credit too, but I think most people believe Sony handed them the early win.

Now what's the case this gen? Microsoft/Nintendo screwed up and Sony is winning, but who is primarily responsible for the PS4's success?

Has Sony done good with the PS4, or have they just not screwed up unlike the others?

It's always both of those reasons.

Last gen MS did a decent start, Sony screwed, 360 flourishes.
This gen, MS screwed, Sony had a good start (but software needs to start coming on time), PS4 is of to a great start.

Credit where credit is due, Sony fixed most of the issues that the PS3 had with the PS4 such as making it cheap at launch and easy to program for, they also didn't really create any new major problems. So the PS4 was pretty much bound to sell well because of this, but because MS and Nintendo dropped the ball with their consoles it probably pushed a significant amount of people to the PS4.

It's a case of both. If the game industry was a sport, it would be golf rather than football or a high scoring sport.

It's about playing the game with the least amount of mistakes ... right now, Sony is making the fewest mistakes, MS has made a few critical ones, Nintendo can't even walk straight without tripping over their feet, they are their own worst enemy. 

The 2 most important reasons of PS4's success:

- Price, very very good
- PS3's legacy

That's an extremely good question ... Had the PS1 and PS2 would have been bad consoles much like the saturn and gamecube would Sony have succeeded ? You see it's not always about the competition screwing up but if your console is just as bad as the others it would gain absolutely no traction either as we know it. The screw ups on nintendo's and microsoft's part are theirs and theirs only but profiting off of their misfortunes means little when your console isn't any good. I would say that it's mostly Sony that has done good with the PS4. If you equate not screwing up as good then you can say that microsoft has done decent with the second iteration of xbox.

1. Microsofts DRM / Nintendos failure to get 3rd party.
2. PS4's good price and a track record of having awesome exclusives.

Why would Nintendo's failures have anything to do with PS4's success?

Sony fixed everything with the PS4. It's the perfect console.

Sony was responsible for PS3 sales with great exclusives later in its life. That will happen in time with PS4 also and the track record is certainly responsible for some of PS4's success, but as of right now a large amount of their success is due to decision by MS and Nintendo. The gamepad screen is pretty dumb. More people are going to dislike it than like it. Microsoft's decisions to go with the higher price point with weaker hardware, forced Kinect, and their completely horrible failure of forced digital and DRM, no used 180's etc. were a huge huge misstep. Their initial PR decisions were so bad I think it could even take a year or 2 to get out of their shadow. All that being said Titanfall is AMAZING.  Rememeber the guy telling people to just deal with it and that is what 360 is for etc. etc.  LOL !!!  When you try to f*** someone who doesn't want to be f***** by you aggressively or forcefully, some people will forgive it if you make it up to them, but more people will run screaming and never look back.