I have read most often that objects experimented with in Mie theory are on a scale not much larger than the incident wavelength (usually a fiber of diameter $5$ microns with an incident wavelength of $630$ nm or so). Can Mie theory be used to describe the scattering pattern of an object any size, as long as a sufficient number of terms in the Mie solution can be calculated?
There is no theoretical upper limit.
The question is whether the description has any practical use. Real-world objects will have some small deviations from the perfect sphere or cylinder shape, for which Mie theory applies.
Look at the polar diagram of scattering of red light from a 10 micron water droplet. Figure 2 in http://www.philiplaven.com/mieplot.htm.
It already has a lot of fringes (compared to smaller objects), and a minor change of size (or shape) would strongly change the pattern. If you're intested to study that you can use the software on the website.