When the limit binds, the Treasury uses extraordinary measures and drains the treasury-general-account rather than issuing net new paper. Bill supply collapses, which pushes short bill yields down and pushes cash into the overnight-reverse-repo-facility.
Bills maturing right around the projected exhaustion date trade cheap, because a few investors will not hold anything with even a small delay risk. Once the limit is raised, the Treasury rebuilds its cash balance with a flood of bill issuance, which drains reserves and lifts short yields again.
Example: a bill maturing just after the projected X-date yields 5.65% while bills maturing two weeks earlier and two weeks later yield 5.30%. That 35 basis point kink is the market pricing timing risk, not credit risk.
Related: treasury-bill, treasury-general-account, overnight-reverse-repo-facility, quarterly-refunding, bank-reserves