I'm not a scientist or medical professional, but I am a nerd! Here's how I understand it:
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Your immune system is both the first line of defense in your body that tries to keep things out and the shock troops that will kill things that slipped past.
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Getting sick is a sign that something made it past your immune system, but getting better is a sign that your immune system has started to kick the intruders out. Your immune system learns by exposure (this is how vaccines work, by introducing a weakened or killed version of a virus to your body, so your immune system can get to know it and develop defenses without you getting sick.)
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So, yes, your immune system is stronger after getting sick, although it's not the getting sick that's important, it's just the exposure to the bug.
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For instance: our baby was recently sick for the first time, with roseola ("sixth disease"). Almost all kids have had roseola by the time they enter kindergarten. But almost no adults ever have roseola: it's rare for someone who has already had it to have it again. So your immune system learned to recognize and repel Roseolovirus and it's unlikely to bother you ever again... which is pretty awesome if you think about it!
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As to kids who are sick a lot, a doctor friend is fond of pointing out that infectious-disease specialists LOVE daycare, because, from their perspective, it's such a great opportunity for developing young immune systems to meet-and-greet with lots of bugs. Some kids do seem to have a cold all the time, but if there are no underlying health issues, I'd assume that's just their immune system learning to deal with the world.
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I find this article http://health.howstuffworks.com/human-body/systems/immune/immune-system4.htm pretty easy to read and interesting: I previously did not know that your skin secretes antibacterial stuff. Amazing!