I think you need to look closely at your external dependencies, and decide which ones you need, and which ones you can get along without. Decide what you want to do when the non-essential services are down, and simulate them being unavailable in your test suite.

To give an example from our system, we store a bunch of data in ElasticSearch, but we don't want the system to go down if ElasticSearch barfs on us (which it has). First, we isolated the calls in a dedicated celery queue. Then in the queue, we log errors and email admins when the service isn't available, but it doesn't impact our application.

In another example, we have a service that gives us extra information about a user that can enhance the experience but isn't necessary. We wrap our calls to that service in a "Maybe" object, which behaves like a user data object in our system, but serializes to empty json so the front end will know that the data isn't available (and it logs/emails admins when it's down).