Obama just needs Missouri to break 375 in 2008. We could get more creative though.
2000: Nader does better somehow (Gore picks a running mate more alienating to conservatives and Nader gets a stronger running mate like Congresswoman Cynthia McKinney) and leads to Bush winning bigger. Bush meanwhile picks a running mate with stronger moderate appeal (Tom Ridge maybe). New Mexico, Wisconsin, Iowa, and Oregon all went for Gore by under 0.5%. Maine's 2nd, Minnesota, and Pennsylvania were under 5%. Maine, Michigan and Washington were under 6%. Those states get you to 366 EV. After that the closest states are Maine's 1st and Vermont (both under 10%) and that gets you to 370.
2004: Polling about a Kerry-McCain ticket had Kerry up 15 points. Alternatively, get a Gore-McCain ticket, as it has the advantage of democratic excitement (he was the legitimate president in their eyes after all) and the crossover factor.
2008: Obama barely lost Montana and Missouri. Meanwhile Georgia, South Dakota, Arizona, North Dakota, South Carolina, and Nebraska's first district were all lost by under 10 points. Also Alaska was within a 3-point margin before McCain picked Palin. Gingrich meanwhile said that if McCain picked Lieberman, Barr would end up with 15%. That's probably hyperbolic, but McCain would lose a lot of support. Have McCain pick Lieberman, Obama pick a running mate more palatable to conservatives (Nunn would guarantee Georgia for him and mean strong southern appeal) and either give Barr a stronger running mate (Barry Goldwater Jr, Mike Gravel, Vice-Admiral Michael Colley who was on the LNC, or Judge Jim Gray) and you could see Obama breaking 400.
2012: Give Gary Johnson a stronger running mate (Gravel, Colley, Roemer, Goldwater Jr, or somebody else) and have Gingrich or Santorum win the Republican primary (or worse - have the speculated Gingrich-Santorum unity ticket go through). GOP ticket collapses and Obama wins NC, Nebraska's Second, Georgia, and Arizona, getting 375 exactly.
2016: Have any Republican not named Donald Trump, Ted Cruz, or Ben Carson win the Republican nomination and face down Hillary Clinton. Alternatively, have Joe Biden or some other mainstream not-Hillary (sorry Bernie, but that Colorado single-payer referendum's results did not bode well for you) candidate be Trump's opponent. OR, have Trump pick Christie over Pence, not rally the GOP coalition (Pence was a great pick given where Trump was weak) and collapse as social conservatives, free marketeers, and Republicans who don't like Christie's sleaze stay home.
I think that if Kasich faced down Clinton in 2016, it'd have been a mess for Clinton - however the only way Kasich was going to be nominee would probably require a deal with Cruz for a brokered convention, which might affect how well he does. Still, I can see him winning all those rust belt states that Trump won (his primary supporters were actually the ones most similar to Trump's, fascinatingly) as well as other close states like New Hampshire, Minnesota, Nevada, Maine, Virginia, and Colorado. I think if Kasich managed to get another conservative-who-is-tempermentally-moderate figure on the ticket (Tim Scott's stance on racial disparities in policing and his focus on family-oriented policies would make him strongest and most in-synch with Kasich I think) he'd do very very very well.