This sounds as a deeper-laying problem. The bandwidth needed to record 24-bit at 96 kHz (assuming stereo) would be:
4 bytes per sample x 2 channels x 96,000 = 750 kb per seconds
Normally one channel 24-bit takes 3 bytes in form of data per sample, but as memory are aligned on 32-bit boundaries in a computer I take this into consideration as it's up to the driver how to handle this data from the device - a max-scenario if you will, hence 4 bytes and not just 3 (which otherwise optimally would be 562.5 kb per second).
Nevertheless, this is not a problem of today - we could capture 16-bit (by sound card) data at 48 kHz back in the days of the Amiga in the late 80's and beginning of the 90's with 33 and 40 MHz CPUs and a whopping 2 to 4 mb memory on 50 mb disks (scsi though, which helped) with 1-2mb data-rate. Scaling this to today's more common specs there is left an enormous headroom to deal with audio data and should not, in theory, be a problem even for an old computer.
But not able to examine the computer on-location we can only make a guess. I would look at the driver (or driver-related aspects) for this case to start with. See if there is an update or a known problem with it. The problem could lay with the frequency and a bottleneck with the interface to the computer not being able to put-through data fast enough. Also check your disk drivers and that they use DMA (Direct Memory Access).
See also if you can increase the memory buffers the software uses to record to before dumping the content to disk. They should be of a size "compatible" (dividable) with the size of the disk sectors (f.ex. 2048 kb etc. which gives you a about 3-4 seconds stereo buffer) as this reduce number of writing operations. You can normally record audio with a very low-end system so a little tweaking of buffers could be enough.
If the problem remains, try install a RAM-disk and record to that. A ram-disk is a virtual disk (logic disk to the computer) that uses memory as its storage enabling high speed writing and reading. This way you can eliminate any disk bottlenecks to see if you need to tweak the disk. Just copy the result to disk when recording is done as ram-disk aren't permanent storage.
If it turns out to be the disk you can defrag it as the number one task. If you have the chance, also increase its sector (cluster for the puritan) size (use 16kb, this removes compacting capability, but that is anyways useless in this context). But find articles on how to do this if you don't already know, as you need to backup your data first.
There can be many other reasons that serves as good guesses incl. the software you used itself, but as said, without being able to examine the computer and see how it behaves, elimination is the "only" method at our disposal :) Good luck.