"Probable cause hearing" may refer to a preliminary hearing that happens well after the filing of charges, at which the court hears testimony in order to determine whether it's more likely than not that the defendant committed the alleged crimes. If the court finds "probable cause," then the case may proceed to trial.
But "probable cause hearing" typically refers to a quicker proceeding involving a determination that there was a valid basis for arrest—that determination allows the authorities to continue to confine a defendant who hasn't bailed out of jail or been released on "OR." This kind of probable cause hearing frequently occurs in conjunction with an arraignment or initial appearance.
The Fourth Amendment requires courts to confirm that an arrest is supported by probable cause either before or shortly after officers take a suspect into custody. A judge or magistrate's signing an arrest warrant serves this purpose, but most arrests don't involve warrants. As a result, officers have to get quick probable cause determinations in order to hold most arrestees in jail. But states don't have to afford defendants an actual "hearing"—rather, it suffices for the prosecution or arresting agency to provide a prompt written statement that a judge or magistrate endorses. (See, for example, D.C. Super. Ct. R. Crim. P. 5(c).)
As a general rule, a probable cause determination within 48 hours of arrest satisfies the Fourth Amendment. (County of Riverside v. McLaughlin, 500 U.S. 44 (1991).) Some states shorten the window, requiring a probable cause hearing within 24 hours of arrest. But if authorities delay a probable cause determination—even one that's held within the requisite period—for an improper purpose, then a Fourth Amendment violation may result. For example, delay is unlawful if it's for the purpose of gathering evidence to justify the arrest.
If a probable cause determination doesn't occur within the prescribed timeframe, the authorities generally have to release a jailed suspect. But courts will often sanction a delay when law enforcement can prove an extraordinary cause for it. (Jenkins v. Chief Justice of Dist. Court Dep't, 416 Mass. 221 (1993).)
Ultimately, unreasonable probable-cause-hearing delays can be difficult to remedy. A defendant must typically prove some kind of harm from the delay—for instance, a confession to a crime after the point at which the probable cause determination was supposed to have occurred. In that instance, the court might suppress the confession. (People v. Nicholas, 218 Ill. 2d 104 (2005).) In other cases, there often isn't much a court will do to address the violation.