The Obama examples have been provided, but I'll give you a little more, if I can!

According to an article in the Washington Post, September 10, 1941 edition, Roosevelt "had tears in his eyes" at his mother's funeral. It doesn't say whether he cried, but it does say he "blinked the tears away".

The Daily Telegraph, in 1994 (March 8, 1994, so within 20 years!), reported that Clinton seemed "close to tears" as he defended his wife over charges of misconduct in the Whitewater investigation. Doesn't seem he cried, though!

But I did manage to find a picture of George W. Bush crying, that is attributed to January of 2007.

A New York Times editorial in 1989 says this:

According to Dr. Cogswell, Washington himself ''was obliged to wipe his eyes several times.'' It was the dramatic climax of the struggle for independence - the hero, the crowds, the new nation up and ready.

Dr. Cogswell apparently wrote this the night of (or the day after) Washington was sworn in to become the first President of the United States. I haven't found more sources than that on Washington crying, though, besides another article in "The Nation" in 1993 using the same quote.

It was written by Teddy White that Kennedy "had tears in his eyes" the night he was elected in 1960, according to this "The Nation" article, titled: Big Pols Don't Cry (Except the Tough Guys).

President Clinton is said to have cried at the swearing-in of Justice Ginsburg.

So yeah, there are records of near-crying or crying!