The short answer: Yes. Over the course of 21 years of trying cases to juries, I have settled a handful of cases during trial. One was a trucking case where a driver failed to set the brakes at a loading dock and injured a forklift or "tow motor" operator who was injured when the truck drifted while the tow motor operator was loading it. Another was a "premises liability" case where a tradesman fell through a set of temporary steps defectively built on a residential housing worksite. And yet another was a medical malpractice trial that had lasted two weeks.
In each case, the defendants initially denied responsibility for what happened. But once the trial gets underway, sometimes a key witness can give powerful testimony, or an opposing witness can perform poorly, or say something downright stupid. Sometimes opposing counsel will overstep his or her bounds and needlessly attack a witness or an expert. And sometimes the judge's ruling admitting or excluding a key piece of evidence will affect one side's ability to present their case. Any one of these things can cause a party to the lawsuit to re-elavuate and possibly change its negotiating position and make a new offer during trial and before the jury renders its decision.
To borrow a sports analogy, a case can gather momentum like a football team driving down the field into the red zone, and this is often the impetus for resolving the case during trial. One thing is certain about a settlement during trial: it produces finality, and eliminates the risk associated with a jury verdict, where there is only one winner and one loser.