In the world of asynchronous programming with Python’s asyncio library, sometimes you need to set a limit on how long your program should wait before giving up and moving on to other tasks. This is where timeouts come in handy. Here are some best practices for handling them:

1) Use the `asyncio.TimeoutError` exception to catch timeouts instead of relying on try-except blocks with a generic TimeoutException. This makes your code more readable and easier to debug.

2) Set reasonable timeout values based on the specific use case. Don’t set them too low or you risk missing out on valuable data, but don’t set them too high either or you could be wasting resources waiting for something that may never come through.

3) Use a context manager to ensure your timeouts are properly cleaned up and released when they expire. This helps prevent resource leaks and memory issues in your code.

Now, cancellations. Sometimes you need to cancel an ongoing task or operation for various reasons (e.g., user input, system errors). Here are some best practices for handling them:

1) Use the `asyncio.CancelledError` exception to catch cancellations instead of relying on try-except blocks with a generic CancelException. This makes your code more readable and easier to debug.

2) Set up cancellation handlers that gracefully stop ongoing tasks or operations when they receive a cancel signal. This helps prevent resource leaks, memory issues, and other potential problems in your code.

3) Use the `asyncio.Task` object to manage cancellations more easily. You can create tasks with specific cancellation flags and then use the `cancel()` method to stop them when needed.

And there you have it! Some best practices for handling timeouts and cancellations in Python’s asyncio library. Remember, always keep your code clean, readable, and maintainable.