Absolutely you can get bored - but you have the creativity to change this. When I start to feel bored with a client, that's a wake up call to me. It usually means I've allowed the same type of talking or self-pity to go on for too long. I have to change it up! Get she or he to see a different perspective.
It's always on me to create my life as exciting or as boring....and it can go either way if I'm not conscious of my choices. As a healer and coach and Phd Psychology researcher, it's on me to change things up and make it fun for me and the client.
When I get bored -- I get my clients and myself moving. We dance out our issues - or draw a piece of artwork - meditate - sing and play in the trees. There are so many ways to help heal someone, that when you start to doze off --- remember to get curious and change the dynamics.
My advice - come at everything with such curiosity, like a little kid again, that there's no room for boredom!
Hope this helped.