This would not be uniform for all courts, even in a single state, nor would it be the same for all kinds of evidence.
Also, few rules regarding the timing of disclosure of evidence are absolute. A judge's handling of a request to introduce late disclosed evidence is reviewed for abuse of discretion, not de novo, on appeal, so a judge is afforded some leeway in how the general rules are applied.
In General Jurisdiction Courts
In courts of general jurisdiction (i.e. those that handle large claims), disclosure of evidence well in advance of trial is the norm. The two big deadlines are usually the close of discovery (a deadline usually set in the pre-trial process), and to a lesser extent, the deadline for disclosing a final set of witnesses and exhibits shortly before trial in immediate pre-trial disclosures. (There are parallel deadlines at different steps in the process for adding new parties to a lawsuit and for raising new legal issues.)
There are two main exceptions to that principle.
Impeachment, Rebuttal And Authentication Evidence
One is that evidence for impeachment of someone else's testimony or rebuttal to something offered by the other side, especially if the statement to be impeached or rebutted was unexpected (e.g. when the testimony to be impeached or rebutted comes from a third-party witness not in close coordination with either party), is generally allowed even in the absence of disclosure.
Late disclosure of evidence solely necessary to authenticate evidence that has already been disclosed is likewise usually treated very leniently.
The Good Cause Exception
The other is that late introduction of evidence is usually allowed only for "good cause" (e.g. the evidence was only discovered or only came into being at the last minute). And, in "good cause" cases, often the other side will be given the opportunity to continue the hearing or trial to allow them additional time to prepare to respond to the new evidence (an opportunity that most litigants will waive in practice to avoid the long delay associated with getting a new trial date).
An analysis of "good cause" also considers the prejudice to each side of excluding the evidence and the materiality of the evidence.
For example, if the late evidence is something created by the defendant, even if it wasn't formally disclosed, a judge is much more likely to allow it to come in than a judge is if the other party has never seen the evidence before. Similarly, a judge will often be quite lenient about late disclose of exhibits that merely compile or summarize evidence that has already been disclosed because it is not prejudicial and could save time.
Judges are also loath to exclude highly material "smoking gun" evidence that would otherwise be admissible and clearly resolve the case on the merits, but are likely to rebuff efforts to introduce evidence not disclosed in a timely motion if it is cumulative or secondary in importance or addresses only a satellite or collateral issue or credibility in general.
A judge is more likely to admit late disclosed evidence if the other party was in some way responsible for its late discovery (e.g. by excluding it from an index in a case where the volume of documents disclosed was voluminous, or encouraging a third-party not to disclose it).
Judges are less lenient towards late disclosed expert witness testimony than they are to admit late disclosed fact witness testimony.
Limited Jurisdiction Courts and Special Proceedings
In courts of limited jurisdiction (especially small claims court), in contrast, there is often little or no pre-trial disclosure of evidence and new evidence can be presented at any time until you "close" your case (either because you say you are done presenting evidence or because the judge limits the time you have to present evidence and cuts you off).
Requirements to disclose evidence in advance also tend to be more lenient in specialized proceedings (e.g. probate disputes, evictions, foreclosure, mental health proceedings) and is usually absent in evidentiary hearing of pre-trial relief (like evidentiary hearings to determine if a court has jurisdiction, temporary restraining orders and pre-judgment attachments) in cases in general jurisdiction courts.
Post-Trial Relief Proceeding Exceptions
It is sometimes possible to introduce newly discovered evidence as late as a motion for post-trial relief (or even in an even later motion to set aside a judgment), but usually, the very latest point at which you can introduce evidence is the close of your case at trial (in the case of a plaintiff, often at the halfway point in a trial as a plaintiff is not always permitted to present new evidence after the defendant closes his or her case).
Consequences Of Evidence Disclosure Deadlines
Evidence disclosure deadlines provide important value by preventing the previously common practice of "trial by surprise" that often led to unjust results.
But, the idea that there is a time limit on disclosing evidence is quite counterintuitive for the average person because it is contrary to how people usually prove things to other people in every day life which is more of a dialog and less of a one sided presentation.
Failing to follow these requirements, or not realizing their importance and finality, is one of the most common ways that people not represented by counsel make mistakes that impair their chance of prevailing at trial.
Many litigants who are unfamiliar with the courts have a very hard time understanding that there is a deadline for disclosing evidence prior to trial, and often, even in trial, will say things like "I have this evidence that I could show you to show that I'm telling the truth" when in reality, at trial, it is your last chance to present evidence and you can't supplement your evidence after the fact.