MBC is currently limited to 1,000 variables (independent + dependent). It is also limited to 1,000 tasks per respondent. There is currently no limit to the number of categories per variable, or the number of respondents.
I am curious about your question, as having 5,000 or 10,000 variables is uncommon. Most MBC datasets have fewer than 100 variables.
Can you describe your model in more detail?