multiple simultaneous email client issues?

I was always taught to NEVER connect to an email server with more than one connection at one time. With the advent of my first smart phone, it checks every 15 minutes for email, all the time. I also log in with a Thunderbird email client or via Google Mail. I think I saw some trip ups but nothing catastrophic.

Is it safe to leave the 15 minute checking on or do I need to or should turn it off every time I log in via a desktop email client????

Inquiring minds need to know.

Thanks!