Northern Piper has it. There is no backlog. The vast majority of cases they reject are rejected on the basis that no sufficiently important point emerges from them. They are disposed of on the papers (in the US). Other countries have brief hearings to give the parties a chance to persuade the court that it should take the matter on, but the principle is the same. By deciding not to hear the point, the court is rejecting the appeal.
The point is made that some countries have several ultimate courts of appeal for different purposes, and there has been criticism of this upthread (including by me).
Captain Amazing observes that there is often a separation between appellate courts hearing criminal as opposed to civil matters. That is not a problem because the distinction between the two is tolerably easy to define by reference to the procedural methods by which the case came to be tried in the first place.
It is a deeper problem to decide whether Miranda (for example) is a criminal case or a constitutional case. It clearly is both, and the distinction is vastly more difficult to draw. And that is a trivial example of the complexity of drawing the boundaries in cases which could conceptually go to either a constitutional ultimate court or a general ultimate court.
Uncertainty about where to go then arises. This encourages people to file in both courts to preserve their position. But what if both courts claim jurisdiction? What if neither does?
It is important to understand procedurally just what appeals are. They are (generally) not merely rehearings of the whole case on the merits, as if it was a trial all over again. By whatever formulation is used in various jurisdictions, intermediate courts of appeal pay great deference to the decision of the jury (or factfinder) who has had the benefit of actually seeing the witnesses. This is considered to be a particular advantage in determining who among the witnesses is telling the truth, and so appellate courts are loathe to intervene in those sorts of determinations.
If appeals were rehearings from scratch, there would be no point in having the primary court in the first place. You may as well just start the process with the court of appeal.
Thus, there are practical restrictions on what an intermediate court of appeal can consider. They tend to have a broad remit, but in practice cases before them tend to resolve to technical questions about the trial judge’s directions to the jury, about whether defences were properly described in the directions to the jury, whether evidence was properly admitted or excluded, and so forth. In some cases, the court will consider whether there really was sufficient evidence for any jury to convict (or find for the successful party), but (again generally) there is constitutional role for the jury and an appellate court does not merely substitute its judgment for that of the jury because it feels like it.
The next level of appeal (to the ultimate court of appeal) is generally only about major issues of law of sufficient public importance to justify intervention. An assertion that an accused is innocent is not nearly sufficient to get the attention of an ultimate court of appeal other than in wholly exceptional circumstances.
In the US, there are various habeas corpus proceedings invented historically so that dubious decisions in what were once thought to be dodgy “Roy Bean” state courts could be reviewed in a federal court and there are appellate structures with the capacity to get constitutional points cranked up in a variety of ways, but it is not at all clear what a whole new level of courts (as has been suggested) would add.
There is no point to injecting a new level of court into the hierarchy just to reduce workload. To do that, you appoint more judges at the overworked level. What would a new level do that was different from the one above or below it? Until that is defined clearly, there is simply no point to a new level if it just has the same remit as the one below it. Once again, why bother with the one below it? You may as well cut out the middle man.