Obviously no. I've been in an environment where I was expected to be breaking my concentration to check my email every 15 minutes and, yes, it was miserable. But that is not what this email signature is suggesting. Four days of silence is ridiculous.

I usually just scan through my email for anything important while switching tasks. If there's something time sensitive or trivial, respond immediately. Otherwise, I put a response on my to do list and get back to them usually later that day. Gmail also has a feature to "snooze" an email to show up at a later time. And of course email filtering helps keep the clutter down.