So given any binary string B:
$$b_1 b_2 \dots b_n$$ $$b_i \in \{0,1\}$$
It would seem it is always possible to make a partitioning of B:
$$ b_1 b_2 \dots b_{p_1}|b_{p_1 + 1}b_{p_1 + 2}\dots b_{p_2}|\dots|b_{p_m + 1}b_{p_m + 2}\dots b_n$$ $$= P_0 | P_1 |\dots|P_m$$
such that when $P_i$ is interpreted as the binary representation of an integer then:
$$\exists{k}:\sum_{i=0}^{m}P_i = 2^k$$
For example here is the first few:
1 = 1 10 = 2 1|1 = 2 100 = 4 1|01 = 2 11|1 = 4 1000 = 8 1|00|1 = 2 10|10 = 4 1|0|11 = 4 ...
Additional examples are here: http://pastebin.com/3B2P4asC
How can we prove this for all binary strings?