I can't recall specifics right now, but in the old Clan Wolf and/or Clan Jade Falcon sourcebooks, during Operation Revival there were one to three occasions where the Clans attacked and conquered their Inner Sphere world target quite easily, crushing some planetary militia with few losses, then saw their warriors brought low by some local disease or illness that was common on that planet but that the Clans were unprepared for. Which in turn put that Clan unit out of commission for a while. As you can imagine this was seen as ironic.
So yes, Trueborns get sick.
Even if you discount them being trueborns though, think about what they generally are. Humans in prime physical condition, excluding those who were prone to illness as a child (I imagine it's difficult to test out of the sibko and become a warrior if you are particularly prone to being sick all the time, that trueborn cadet with a weak immune system, or who develops a lot of childhood medical complications probably gets relegated to a lower caste), in that 20-40 age range.
In that age 20-40 stage of human existence, you have an above average chance of getting through with fewer diseases and health problems and where, when you do get sick, you tend to bounce back faster. Heck you even bounce back from things like sleep disruption or hangovers faster. Your body can also handle a lot more poor nutrition choices like fast food quite easily. Somewhere in your 30s or 40s (it varies by person) age starts to slow you down and the risk factors of certain health conditions begin to increase. Your body's immune system can weaken, hormone levels tend to start changing, and so on. In your 20s you could go to that all-night party, throw yourself on a couch, and then go for a run the next day and probably feel mostly fine, you bounce back fast. In your 40s you do the same thing and it feels like you need DAYS to recover.
There are numerous exceptions to this, but that holds true for a lot of potential health issues and complications.
So again, we're talking about very healthy human specimens in peak physical condition. Akin to real world professional athletes who made the cut out of hundreds or even thousands of possible candidates. That has way more to do with it than them being trueborn.
It should be noted that we also know Clan medical science is pretty advanced. So that works in their favor too. The average Clan Trueborn warrior, age 20-35 is going to want for nothing. Medical treatments, therapeutics, preventatives, regrown limbs if viable, advanced cybernetics otherwise. The Clan will spare no expense on them. That's one of the advantages of being the leadership caste in a caste-based society. It's only once they reach solahma status without a bloodname that they start to be seen as lesser-than-deserving of the best available medical support.