In either your example or the bicyclists situation, there is nothing to say that you are absolutely responsible in any way for what happens. That is, you're not negligent as a matter of law in those situations. But what Texas law permits is for a jury to consider the possibility that you might have been negligent, as a matter of fact, in some fashion. The jury can say you have no responsibility; it can say that you are entirely responsible; or it can say that you share responsibility in some proportion to the defendant.
Functionally, this serves to reduce the defendants' liability by the percentage of your responsibility. Say, hypothetically, a jury in this case finds that the defendant is 75% responsible for the accident and that the bicyclists were 25% responsible and awards $1,000,000 in damages. All other things being equal, the defendant would bear liability only for $750,000 (75% of $1,000,000) of the damages.
There obviously has to be some evidence to support a finding that you had any responsibility for the incident that caused your injuries. But if there is some evidence, we trust juries to allocate blame among the parties, including the plaintiff(s).
Like I say, in Texas we don't allow an "assumption of the risk" defense, which would be the legal equivalent of the notion of making them responsible as a matter of law for their injuries. But we do allow the defendant to put some of the responsibility on the plaintiffs if there is some proof to support that effort.