We frequently use the BIE plugin to migrate blogs with thousands of blog posts without any issues. However you might see an error message if the importer is failing on your site. 99% of problems are usually caused by underpowered web hosting, WordPress theme you are using, or having too many WordPress plugins installed. When using some hosts, they will slow down your site if you use too many plugins or a slow theme.
We use Bluehost for hosting our sites and client sites and have not experienced any issues yet.
Here are some tips for easier migrations:
1. Activate the default "Twenty Nineteen" or "Twenty Fifteen" theme. You can do this from the "Appearance" page in your WordPress dashboard.
2. Deactivate all plugins except for Blogger Importer Extended. You can do this from the "Plugins" page in your WordPress dashboard.
3. Make sure your host is using PHP version 7.3 or higher. More info here.
4. Increase the PHP Memory Limit to 256MB via this guide. Your web host will be able to assist with this.
If you need any help with the BIE plugin you are welcome to email email@example.com and we can assist you directly.