It is often the case that a seat appears to need nothing more than fresh woven material. However, removing the old material often testifies to the strength that even a fairly tattered seat can still yield. Handle the skeletal frame of the piece and you might find that there are greater issues. Often a single joint has failed or an old stopgap fix from a previous owner suddenly surfaces in previously obscured locations or it might just be that the whole thing just needs to be taken apart and freshly glued up before being rewoven. Below are a couple of examples of repairs that followed previous stopgap measures. One quick tip on locating areas of concern:
Look for things that might seem out of place (though qualify all considerations with the possibility that any conclusions you draw could be purely subjective). This could be a screw that seems to have no companion elsewhere in the chair, a slight change in stain/paint color, or a sudden shift in grain pattern that doesn't look original.
Here's a barley twist chair that looked ok with the old, broken cane in it. Seeing the caned seat can distract from what issues lie beneath it. When in the back of a garage it looks like a solid chair. Flip it over or remove the cane and you see a repair that also highlights a weakness in the chair's design that has clearly caused problems in the past and likely will again.
These little tenons on the side leave very little meat above them on the rear piece, which also has the most stressed section of cane given a deceptively large size and depth of the chair (104 holes) and the angle of the seat back. There were breaks on both sides in this same spot and the rear was tilting forward considerably, tugged by the cane and held in place by just a few discreetly placed wire nails. Cane, even when the pattern has broken in spots, is strong enough to stress such a design to its breaking point. In this case both the strip meant to hold the cane in the rear broke and the side "joints." At this point, cost becomes prohibitive to really do what the chair would ideally need. Just caning a seat of 104 holes is very expensive. The cheapest fix here was to merely replace the rear piece with a compression fit and drive sufficiently large screws in from the bottom to hold the new piece in place. This might have involved something more elaborate but these are the kinds of choices you have to make when you find unanticipated flaws in a chair you paid little for and hoped to avoid putting too much into thereafter.