Is it possible that part of sequence is more complex than all sequence because the best way to encode it is to use the complete sequence and starting and ending positions of the fragment.
Maybe, for example, string of hexadecimal digits of $\pi$. Or something else.
I am not sure, but I would say yes, however, only by $O(|n|)$ (the length of program/TM that would truncate such sequence including the desired length of output sequence). An example of such sequence might be some prefixes of output of Busy Beavers. Still, this is just my intuition.
I hope it helps ;-)