Hard to answer without example data and values. But I can see this happening if there is a lot of data and/or there are a lot of correlated variables.

Might be useful to examine subsets of the variables to test specific hypothesis. These issues are discussed widely e.g

https://doi.org/10.1111/j.1365-2699.2006.01584.x

Try univariate sensitivity testing?

Just adding the following recently published article which seems to be both timely and relevant:

Article Testing the ability of species distribution models to infer ...

Thanks Darren Norris for this reference and the previous suggestion.

By the way, although there was not high correlation between the initial set of variables we used, we tried some subsets of variables based on a number of more specific hypotheses and we found some combinations that essentially maintained TSS values, but identified at least one variable of much higher importance values (>0.2).