Not a silly question at all, and one that has been hotly debated in the scientific community.
Some anthropologists have supported the proposition that Neanderthals interbred with early modern humans in Europe, and even trace some European traits to that interbreeding. However, genetic evidence has made it increasingly probably that all extant humans are descended from populations that emigrated from Africa, and did not interbreed with local populations of earlier types of humans.
Mitochondrial (non-nuclear) DNA has been extracted from Neanderthal bones, and has been found to be very distinct from that of modern humans. It is on this basis that an increasing consensus has developed that Neaderthals be regarded as a full species relative to Homo sapiens, rather than a subspecies. However, there is still some dispute about this, and it is possible that analysis of nuclear DNA could tell a different story.
The issue of interbreeding between humans, Klingons, or other aliens is completely different. Aliens would not share any common ancestry with humans (whereas Neaderthals would have had fairly recent common ancestry. Having developed separately, the genetic code of aliens would be completely different, making it impossible for interbreeding to take place. It would be like shuffling together the pages of two novels written in English and Chinese and expecting it to make sense.
Actually, the consensus was that, although unlikely, it would not necessarily be impossible. However, it would take experiments to find out.
It is quite possible that modern humans and Neaderthals would have been able to produce hybrids, either sterile or perhaps even fertile. However, the evidence suggests that if interbreeding did occur, it was not widespread enough to leave a trace in modern populations.