Subset partition problem variant
For its complexity, note $N$ the sum of all your integers. Then, the number partitioning problem injects itself into your problem with $K=\frac{N}{2}$

I found this paper which deals with almost your problem. It aims to maximise the number of bins used, but instead of working with each partition must have a sum of at least $K$, you have each partition cannot be completed with another item.

Edit: Another user already asked almost the same question: https://scicomp.stackexchange.com/questions/8216/bin-packing-maximise-number-of-bins-fukubukuro-problem