In short, yes there is a situation where you could end up owing, but it is extremely rare and with appropriate advice from your attorney, it is an outcome that is usually avoidable as your case progresses..
This is a topic I simultaneously love to discuss with clients because it shows they care about their cases and they're reading their contracts, but also am concerned to discuss because the answer might drive people with legitimate claims away from seeking legal help with their cases. It is a tricky question because it involves attorneys following strict rules of professional conduct about advancing litigation costs; it involves practical, business considerations for attorneys running their law practices; and it can require difficult forecasting about the outcome of cases without knowing all the relevant information.
First, the rules of professional conduct prohibit attorneys from paying litigation costs. The idea is that it gives the attorney too much of a stake in the litigation, and therefore it is just something that cannot be done. However, attorneys are permitted to advance litigation costs, on the promise that they are paid back at the end of the case.
Now, the business determination: lawyers like to have happy clients, and we don't like to be too loose with money. Costs advanced are supposed to be investments in the case, meaning that in the attorney's judgment based on experience, the case has a value worth spending the money on. Further, not only will the money be paid back out of the anticipated settlement or verdict, but we expect there also there will be an attorney fee at the end of the case
Again, that is tricky because the only real measure of a case's value is what a jury will award. Most cases do not go to trial, and as a result we don't know the true value of most cases. Still, through experience, attorneys gain a sense of what a fair settlement value ought to be for a case. Although we can make recommendations, the decision whether to settle the case lies exclusively with the client.
At the end of the day, these judgments can be wrong, and some cases do not settle. Imagine a scenario where you have a lot of back and neck pain after a modest car accident, you get medical treatment, and the other driver's insurance company only offers half of your medical expenses. They might believe that the impact was too minor to have caused the injuries, or raise some other common defense we see. After litigation the case ends up in trial, and the jury might agree with the defense's theory. Only now, after probably more than a year and several thousand dollars of costs advanced, you are not only underwater for medical expenses, you are also in debt to your attorney who advanced costs.
Hopefully, your attorney would have advised you along the way of the risk of this happening, the pros and cons of settling versus trial, and allowed you to make an informed decision about how to proceed. Your attorney is ethically required to bill you at the conclusion of your case for the money he or she spent on your case to prepare it for litigation and trial.
To be clear, an outcome where the client ends up owing money instead of making a positive recovery does not happen often, and it can happen for a lot of different reasons. It is a sad reality of the personal injury practice that some injured persons at the end of the day, just do not get a fair shake. Again, though, that is a rarity.