Of the actual Tamagotchi, never. Of the routine that sometimes you tend to lean on as a habit, yes.
But that's why you always have to set a goal within the Tamagotchi, or be completely okay with "fate" taking over the characters and such.
Always have to make sure that you go out and do things with your Tama, in your every day things, as well as just... I don't know. I find the routine of knowing exactly when pooping happens, or exactly when they're going to evolve, etc, takes the fun out of it, since we can't plan those things. For ourselves, friends, family, etc, and I don't think we'd WANT to know; therefore, I find that breaking yourself of the routine of anything like that makes the Tamagotchi far more amusing and fabulous.
D: Unpause it! Or.. if you get bored of playing the tama you have running, see if you can find one you like better. Personally, the only reason I ever get bored with them is because I have nobody to connect with. Try and find out why you get bored of them and maybe then you'll have a better chance of not being bored.
I'm growing tired of my tama-go so I neglect it without making it leave. I can't wait until my tamagotchi set gets here. In the meantime, I'll search my room for my v3, v4 ,v4.5 and v5. When my tamagotchi set gets here or when I find my old ones I'll get new batteries from Batteries Plus and play with them.
I find that after starting up a tama that I haven't in a while, I get bored with in 10 minutes and then start up a new one. But my iD L came yesterday, so I think I might not get bored with that one.