Are there any "secret" rooms that contain cube bits?

OK. I have 31 cubes, and 7 bits. There is one missing. Where in the world is it hidden?

Before you say, "That's impossible to answer without knowing all the other ones you've already collected" there are a few relevant bits of information:

• Going through as if this were my first play-through:
  • Not exploring any secret areas
  • Avoiding all anti-cubes, even the more obvious ones
• With the exception of secret doors:
  • All rooms on my map are gold
  • All non-secret paths have been thoroughly explored

---

_{Why am I going through the game this way? Because it's fun! During my first play-through a few years ago, much was left unexplored since enough stray anti-cubes were collected to finish the game without exploring everything.}

---

Given all that, the real question is:

Is there a secret room somewhere with a cube bit in it?