My guess is that this happens because you are using a standard reader for vtk files from a package.
My recomendation is to write your own reader. It looks that is has a variable length header in text that you should parse and then a binary blob that is the array. (I think the binary version and no compression is better to read random points..)
Let's assume you need 10x0x10 points arround each 1440 measurement, that's only 1,500,000 flonums to read. Instead of the full cube that has like 1000x1000x1000. Beeng too optimistic, that is a 1000 speedup but real life and disk storage are harder, so I guess a 100 speedup is more realistic. So the time will go from 4 minutes to 2 seconds. Also, a small memory footprint may reduce other problems and make everything faster.
I think it can be implemented in less than a week, so I strongly recomend to try it.
(My recomendation is to load only the 10x10x10 datas and then just drop them and load the next 10x10x10 data. In other scenarios it may be helpful to save them in a hash or something and avoid reading them again later. But in this case the simple solution seams to be fast enough and easier to implement.)