Managing limits of $wpdb->get_results

In a plugin I am doing the following as part of exporting data collected by the plugin itself:

$wpdb->get_results("SELECT * FROM $table_name");

Now when the table has only a limited number of rows, there is no problem and I am able to export all of the data and ->get_results() works fine.

However when the dataset grows past a certain point, ->get_results() fails silently and ->print_error() gives no information.

Logically it would seem that I am hitting a limit somewhere and am interested in figuring a way to access all the data (in chunks I would guess)… Can anyone provide any hints to help a newb? 🙂