While testing the fix of MC-3, I found this bug. I then tested it back in 1.8 and it was still there, so I don't know how long it's been around or even if there is a ticket for it already (I tried to look it up but couldn't come up with anything).
Anyways, when you travel to the Nether, generating a new portal, and build a new portal, even just a couple blocks away, and destroy the old one, the overworld portal doesn't adjust your destination coords in the Nether, placing you where the old one was.
To reproduce, build a new Portal in the overworld and travel through, generating a new portal in the Nether. Now build a second portal in the Nether a few blocks away and destroy the first. Now if you travel through the new portal to the overworld and then back again, it will place you where the old portal was, even if it was destroyed. The game also neglects to generate a new portal in that location, so you just spawn in with no portal blocks touching you.
Related issues
discovered while testing
is duplicated by
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I would add that it doesn't seem it work for more than a few blocks away but It is definitely present in 1.8.1 Pre-1.
Edit: Seems to only work some of the time.
Edit 2: This bug (feature?) is based on what Sol Toder said below. If you wait 60 seconds this doesn't occur. This may be working as intended.

According to the wiki, nether portals remember the destination from the last time they were used for 60 seconds. So this is less of a bug, and more of a poorly thought out feature.

Well if a [Mod] or member of Mojang wants to come mark this poorly thought out "Feature" as "Works as intended", then they can go right ahead. I just think it should be adjusted so it doesn't feel so buggy (spawning in a location without portal blocks touching you). At the very least, they could make a new portal generate where the old one was if it's no longer there.

WAI for now. Right now we remember portal destinations for 30 seconds after they're last used, because it's really expensive to do otherwise.

Excuse my ignorance, why can't you "forget" portal destinations when a portal is built or destroyed?

Scanning for them is resource-intensive.
Technically a duplicate of MC-72224, but this is more detailed.