Are there any two triangular numbers that add up to a perfect cube?

Yes. There are infinitely many solutions. Let $x_1=k^3$ and $x_2=k^3-1$. Then $$ \frac{x_1(x_1+1)}2+\frac{x_2(x_2+1)}2=k^6=(k^2)^3. $$

Yes.$$(1+2+3)+(1+2+3+4+5+6)=6+21=3^3$$ Another example is $$6+210=6^3.$$