Competition for the most elite law students is ratcheting up, with several national law firms boosting starting salaries for recent law school graduates to $190,000 in recent days.

But whether the raises will lead to an across-the-board pay bump—as they have historically—is an open question. Compared with past rounds of pay hikes, fewer firms have rushed to match the new pace, set last week by New York firm Milbank, Tweed, Hadley & McCloy LLP.

Milbank...