It depends on how the subsampling is done. If it was random, it is not surprise at all that you get different assemblies as low-abundance groups may be mostly or completely missing from subsamples. If you do a digital normalization subsampling (see khmer) which preserves DNA sequences with low reads coverage, then you can expect to get a similar assembly.
Depending on the average depth in your full sample, it is possible that sequencing errors overwhelm the assembler's ability to pull DNA fragments together. It is not a matter of overloading, but simply a signal being drowned enough by noise. If the average assembly is >500-1000x, and especially if it is in many thousands, that will create an artificial fragmentation of resulting DNA contigs. Subsampling the reads down to 20-100x will potentially solve this problem, but not if the subsampling is random.
I think a more precise description of what you did will likely yield better answers.