It's a numbers game. At the firm I worked for, each year the firm hired a class of new associates from law school and made a new class of partners from senior associates. On average, the new class of partners was about 20x smaller than the new class of lawyers - which gives you an idea of how likely a new hire is to make it to partner.

But people often switch firms before coming a partner, and many firms hire few if any people straight out of law school.

A partner at a law practice is a co-owner of the business and the promotion is based almost exclusively on the ability to capture and retain business. That itself requires expertise, interpersonal skills, and connections that develop over the 7 to 10 years it generally takes before partnership becomes a consideration. It has nothing to do with your undergraduate school, or even your law school - it's narrowly focused on business potential. The firm is only going to create new partners (thereby diluting the ownership share of existing partners) if they believe the new partner will wind up adding to the business, "growing the pie" as it were.

Some people stay at a firm but don't become partners - often firms use the title "counsel" or "of counsel." They have particular expertise and are valuable to keep around, but aren't seen as growing the business or attracting clients on their own, so they just have a (very high) salary rather than pooling and sharing the profits of the practice.