WMG / 5 Centimeters per Second

Akari's parents turned Takaki in to the police.
For staying out all night with their daughter without permission. He is therefore banned from contacting her ever again. This is why he never writes, calls her or visits her ever again despite her apparently still living in the same place with the same number and address as before. He'd rather get depressed and quit his job and mope than see her because the law won't allow him to.

Takaki and Akari are a duprass.
Hey, nothing in Cat's Cradle says that a duprass's lives can't suck sometimes. Takaki may also be Billy Pilgrim.

Takaki has clinical depression
...And probably had it throughout the entire movie. He moves often as a child and appears to have few friends in Elementary School, so it's likely his depression starts from that. Akari was the first person he feels truly close to. Once she is gone, Takaki's depression gets worse due to the resulting loneliness. This is why he always sits on the hill and stares off into the distance, and why he writes texts but never sends them. Takaki is trapped in a lonely emotional prison; his brain chemistry is probably so off that he's too inwardly-focused to see the genuine feelings that Kanae has for him. Kanae misinterprets this loneliness as Takaki "wanting something beyond what she can offer," when in reality he is looking for nothing more than a way out of his own head.

By the third part of the movie, Takaki quits his job and leaves his girlfriend of 3 years. He likely expected his job and relationship to make him happy, but his depression kept him from really enjoying either; he's still stuck in his own head just like in high school. So, he walks around aimlessly instead. Only after he sees Akari does he start to have a slight bit of hope.

Akari was never actually there at the end.
Seen this in some other places.

The idea is that the Akari that Takaki saw at the end was either a hallucination or an Identical Stranger/stranger he just mistook for her at a fleeting glance. Thus, her not bothering to wait for the train to pass. If it's a hallucination, then his letting it disappear also ties in to his letting go.

In all frankness, even if they were no longer romantically involved, the idea that one wouldn't wait to greet a long-lost old friend... Well, the theory that the woman was not actually her is slightly less depressing than their relationship having fallen so far apart that she won't even wait to exchange greetings as friends.

Supporting this are the lyrics of "One More Time, One More Chance", playing over this scene, which strongly imply that the her he sees is but a fantasy in his head:
I am always searching somewhere for you
The opposite station platform, the back alley window
Even though I know you won't be here
...
I always end up looking somewhere for your smile
At the railway crossing of the express train
Even though I know you won't be here


http://tvtropes.org/pmwiki/pmwiki.php/WMG/FiveCentimetersPerSecond