In principle, MICE should be able to handle large amounts of missing data. Variables with lots of missing data points would be expected to end up with larger error terms than those with fewer missing data points, so your ability to detect significant relations to those variables would be limited accordingly. That's an advantage of having multiple imputations and analyzing results from all of the imputations.
The greater the fraction of (a) cases with missing data or (b) predictors that are frequently missing, the more imputed data sets you will need. See Section 3.10 of Frank Harrell's Regression Modeling Strategies for some "rough guidelines" about how to proceed. If the missing values are for variables that you consider important based on your understanding of the subject matter, he says: "Extreme amount of missing data does not prevent one from using multiple imputation, because alternatives are worse."
More important than a "cutoff" for missing data is to consider carefully (1) the intended use of your model and (2) whether the "missing-at-random" assumptions needed for multiple imputation holds in your case.
In terms of (1) if you, say, intend to use the model for prediction but some variables are inherently hard to get, then there's no sense including them in the model. Also, you should use your knowledge of the subject matter to consider variables for inclusion. If you suspect that only 10 or so will be important based on such knowledge, maybe you should just use those 10.
In terms of (2), "missing at random" means that the probability of missingness doesn't depend on unobserved data. As Stef van Buuren says in Section 1.2 of Flexible Imputation of Missing Data (FIMD):
If the probability of being missing is the same only within groups defined by the observed data, then the data are missing at random.
That's a typical starting point for analysis. van Buuren discusses ways to evaluate the sensitivity of the results to violations of the assumption in Section 9.2 of FIMD.
