"Suspect" is a strong word. We do not know what anyone, except the POV characters, suspect. We can only suspect, or assume that they suspect, unless they voice their suspicions. People might very well suspect things about Ned's bastard son. But the story of his breach of honour in bringing a bastard child home from the war was no doubt an appealing story for people: Why would he say such a thing unless it was true?
We know that he sacrificed his own honour to protect Lyannna's son. Much like Robb chose to sacrifice his own honour to protect the honour of the woman he married, Jeyne Westerling. But to the common man, it would be a plausible story that an honourable man brings home his bastard and raises him with his trueborn children.
However, GRRM's official "cover story", as I have always understood it, is what Littlefinger says to Sansa: It is rude to pry into people's bastards. It is described in this conversation between him and Sansa, when they decide on the best cover story for Sansa when she comes to the Eyrie.
(Sansa) "Couldn't I be the trueborn daughter of some knight in your service?
Perhaps he died gallantly in the battle, and..."
(Littlefinger) "I have no gallant knights in my service, Alayne. Such a tale would
draw unwanted questions as a corpse draws crows. It is rude to pry
into the origins of a man's natural children, however."
As for why no one would 1) wonder how Lyanna died, 2) connect it with Ned bringing home an infant son, which seems a pretty obvious connection to make, from our perspective, one has to take into account that this was war. Momentous events took place, the Targaryen dynasty fell, many great lords died or were exiled, many knights were killed, many "war babies" were no doubt born. Not all people are as inquisitive as someone like, for example, Littlefinger or Lord Varys.
And also, of course, it fits the story much better this way. :)