For a real-life example look into dengue fever. There are multiple variants that - to our immune system - look very similar, but antibodies for one do not protect against the others. This means that if you recover from one variant and are subsequently infected with a second you can have a really really bad time.

The reason the second infection is so bad is exactly as outlined above: your immune system 'recognises' the second variant and produces antibodies that it learnt when fighting off the first variant you recovered from. Unfortunately, these antibodies are useless against the second variant, and you have a bad time.

This exact same issue happens with the current vaccine for dengue, which is why it is only given to people who have already had dengue of some kind, or in populations where most people have had it before. For a situation where this did not happen see the section titled 'Phillipines controversy' in the wikipedia article.

There is currently a lot of work being done to produce vaccines that protect against all 4 variants at the same time, with some promise for the years ahead as trial results come in!

https://en.wikipedia.org/wiki/Dengue_vaccine

Your immune system is not intelligent. Through natural selection it has evolved to be able to handle some things, but it can't see permutations like that, as they are not something it evolved for, it's all chemical/physical matching.