The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter?
1. **Identify the letters and their frequencies:**
   The phrase "COLORFUL TARTAN" consists of the following letters with their respective frequencies:
   - C: 1
   - O: 2
   - L: 2
   - R: 2
   - F: 1
   - U: 1
   - T: 2
   - A: 2
   - N: 1

2. **Apply the constraint:**
   The problem states that neither bag can contain more than one of the same letter. Therefore, for letters that appear more than once (O, L, R, T, A), each bag must contain exactly one of each of these letters.

3. **Distribute the letters that appear more than once:**
   - O: One O in each bag.
   - L: One L in each bag.
   - R: One R in each bag.
   - T: One T in each bag.
   - A: One A in each bag.

4. **Distribute the remaining letters:**
   The remaining letters are C, F, U, and N, each of which appears only once. Each of these letters can be placed in either of the two bags. Since there are 4 such letters and each letter has 2 choices (either bag), the total number of ways to distribute these letters is:
   \[
   2^4 = 16
   \]

5. **Calculate the total number of ways:**
   Since the distribution of the letters that appear more than once is fixed (one in each bag), the total number of ways to distribute all the letters is determined solely by the distribution of the letters that appear only once.

Conclusion:
\[
\boxed{16}
\]