Michael D. Cohen, President Trump’s former fixer, always had a high self-regard for his ability to talk — or bully — his way out of challenging situations, whether acting on his own or on behalf of Mr. Trump.

So when federal prosecutors in the Southern District of New York began investigating Mr. Cohen, he seemed to undertake a brazen and risky legal strategy: offer enough information that it might prompt prosecutors to ask a judge for leniency for him — but nothing more about his or others’ activities.

On Friday, the prosecutors made clear that Mr. Cohen was less useful to their investigation because he would not fully cooperate, therefore he would not reap benefits, such as a government letter on his behalf.

They said Mr. Cohen had refused to sign a full cooperation agreement, the sort most people in the Southern District sign when agreeing to testify against their partners in crime. Under that sort of deal, witnesses must admit to every crime they have committed and offer any details concerning crimes by others, even ones the government did not know about.