Is there a limit to the number of options you can store in one settings field of wp_options?

The limit is probably not WordPress specific, but caused by PHP.

max_input_vars might be set to 1000, so not all fields might reach WordPress.

You can store 4,294,967,295 or 4GB (232 – 1) characters in one option, so I don’t think this is your problem. If it is, you should consider a separate table.