The thing that Blahness is referring to is a bug, but the rest are not. The Blahness issue there was something that was happening in 3.060 betas, which we thought we had stamped completely out, but which apparently still happens on some rare circumstances. We don't yet have a save that will let us demonstrate this, and we haven't seen it ourselves since that was fixed, and reviewing the code did not turn up any obvious culprits, so on that one we're sort of stuck waiting for someone to send us a save from before a CPA emerges and disappears (no threat increase).
But, as I said, that's pretty rare. What's much more common, and by design, is what TheMachineIsSentient is reporting. First off, I think there is some confusion here about how CPAs actually work. This
wiki topic will hopefully clear that up.
Now: that said, and hopefully read, the reason that a CPA might be more focused or more diffuse is hopefully clear. If you are really strung out in the galaxy, CPAs are likely to be the bane of your game, the most challenging thing in it. If you are in certain other types of positions, they may be less of a threat, just logically based on how they work, above. But to win, you have to expand at some point, and that's when CPAs become the most threatening.
But yes, the only immediate effect from a CPA countdown ending is that your Threat meter jumps up by the value of the CPA, and suddenly there are hundreds or thousands of angry AI ships out in the galaxy hunting you... what happens from there will vary heavily from map type, your position, the AI's position, decisions the AI makes, etc. The AI ships will try to clump together, but if they are so spread out at start they never even see their peers, then you're more likely to get a dribble as wyvern83 notes.
Also, to add to wyvern83's explanation of reinforcements (which was excellent and correct), let me also point you to the
wiki topic on that one, which lays it out in immense detail if you want.