I read another question, and its answers, about how vaccines work, but I don't see there, and and don't understand, why some infections can, seemingly, not be immunized against at all. For example, someone who gets the flu (or the flu vaccine) is immunized against that particular strain of flu for life (which was relevant, IIRC, when H1N1 came back a few years ago: many people above a certain age had immunity), whereas one person can get group-Α strep throat over and over again. Why is this?
MCM's answer is quite correct. There are quite a few specific ways various pathogens escape "immunity" (long term, often not truly life-long, protection).
To quickly restate what he has said: First the pathogen can mutate to be different enough that your body won't recognize it (ie different strains of flu or HIV), it can hide in cells or immunoprivileged areas of the body, it can actually shut your immune system down, actively combat your immune system, or prevent an immune response from being mounted.
And just a few more thoughts:
There are certain pathogens we are still trying to figure out how they avoid the immune system (the technical quality is immunogenicity). A good example human respiratory syncytial virus (RSV). Again this is actually an active field of study for a great many pathogens. When we test many of these "tricky" pathogens in animal models, we find that they often can gain immunity.
+1 @MCM, I was trying to cram this into a comment, but I didn't think I could.
Some interesting areas to read up on if you are interested in the subject are immune avoidance/evasion and forward genetics. You will more often hear about reverse genetics, but forward genetic experiments can allow us to see how the "chess game" will be played out by the pathogens in new situations.
The answer is sort of hidden behind the very first paragraph of your linked question:
The body mounts an immune response based upon a particular epitope (molecule/protein chain) derived from the infection itself.
If the bacteria/virus/parasite evolves so that the molecules/proteins on its surface change significantly, then the version stored in Memory Cells will no longer apply. There's some wiggle room depending on epitope length and specificity of the Immune Response (highly specific MHC proteins will invoke a very strong, very efficient response - while less specific [usually B-cell or early T-cell responses] will invoke a weaker, less effective response), but because pathogens can mutate so quickly it's not uncommon that you'll encounter a descendant of a virus you already encountered that has dramatically different characteristics.
Some pathogens are also just very "clever" at avoiding the immune system. Sometimes they can produce shells which can't be broken down to derive epitopes from, sometimes they live within host cells and simply hide (the most well known example being HIV), other times they can present the same "I'm part of the body!" MHC proteins that tells the immune system not to attack them, and they can even hide in parts of the body where the immune system rarely visits (eyes, brain, other very specialized tissues).
And at times the pathogen can be both very adaptive and very "clever", as in the case for Staphylococcus aureas - which can inactivate antibodies through surface proteins and has the ability to adapt to new antibodies very quickly.
You can think of it like a game of chess. If you play against the same person all the time you're going to learn their moves, their strategy, and be able to beat them very quickly. If you switch partners, or your regular partner learns new moves, you'll spend some time adapting to the new play style you've encountered before being able to win consistently again. Bacteria and virii are simply very, very good at coming up with new play styles.