It used to be very common to make partner, but as you say, it isn’t as common as it used to be. When it was common, if an attorney didn’t make partner at his firm after a certain time period, he often felt like he had to leave that firm.
Now, since it isn’t as common to make partner, people don’t feel they have to leave if they don’t. However, that is still a potential response, simply leave. The attorney could seek a lateral position with another firm, where he has a better chance of making partner, or where he won’t feel upset about not making partner. Some lawyers will join together with other experienced attorneys and open their own firm. Others will go in house or seek another job where being a partner is not an issue. Still others will go solo. And of course, there are others who leave the practice of law altogether.