Katherine did, on one occasion, believe she was pregnant. The “I’ve missed my bleeding” scene in the Tudors is accurate (to some extent - she probably didn’t run through the galleries and burst into a council meeting.) However, the following month, her hopes were dashed. Whether or not it was a miscarriage or just a missed period (for some medical reason we cannot diagnose) is unknown. However, we do know that rumours of a pregnancy and of her fertility followed Katherine’s entire time as queen - it even followed her to the scaffold: the possibility of her mothering an illegitimate child whilst queen was a fundamental reason for her execution.
If there was a medical reason for her failure to get pregnant, then it would most likely be on the part of Henry, rather than her. He was 49, and terribly overweight, with countless other health-related problems, including his mental health: it’s easy to imagine that he had problems in the bed chamber. He was definitely sexually active with Katherine, yet was unable to impregnate her, and it’s generally believed that he was impotent.
Of course, there may well have been a chance that Katherine herself was unable to bear children. We cannot be sure, because she also knew how to avoid pregnancy, and because she was also only fully sexually active with two men (for certain.) With Manox, they only got so far as heavy petting, and with Culpeper, both denied doing anything even remotely sexual. Dereham’s relationship with her didn’t last too long, and she would most certainly have been employing her knowledge of Tudor era contraception in that time, whilst, as mentioned above, Henry was probably unable to “perform” in bed. If Katherine was infertile, then we will never know with any degree of certainty, though it is a possibility. As you said, she was definitely young enough to be capable of birthing children, yet she failed to.*
There is a theory that Katherine’s relationship with Culpeper was out of a desire to get pregnant when she could not do so with Henry: this theory, of course, supports the notion that Henry was impotent, but, regardless, Katherine still did not manage to get pregnant, which might suggest her own infertility - but probably not, since getting pregnant is difficult even in the modern day. Personally, I do not agree with this theory at all.
In my view, Katherine’s lack of pregnancy was probably due to Henry’s age and health problems, rather than her being infertile. It is a shame that she never had any issue, for it might have prevented the Lascelles siblings from speaking out against her if she had a child to strengthen her position - but this is all just speculation. Indeed, if she did have a child and her past/relationship with Culpeper was discovered, then there is a possibility that she would have been carted off to the block very quickly for possibly bringing a bastard into the succession, and her child could have suffered a rather nasty fate with her (and, probably Dereham/Culpeper).
It’s an interesting topic to consider though, especially when it is remembered that Katherine Howard was very much a symbol of fertility for Henry, and for the court. Her youth, as you mentioned, and her beauty and vivaciousness made her represent Henry’s own virility. Katherine’s life, and death, therefore, became a rather poetic butchering of all of Henry’s idealism, but that is an essay for another time.
*EDIT: Another point to mention is that Katherine’s mother, Jocasta, had 11 children between her two marriages.