No. Let $x_1,x_2,\dots,x_n$ be the bits of the binary sequence.
Suppose $\Pr[x_{i+1} | x_1,\dots,x_i] = 1/2$ for all $i$. Then it is impossible to predict the next bit; however, it is also easy to see that $x_1,\dots,x_n$ is uniformly distributed on $\{0,1\}^n$, so has entropy $n$ (high entropy; no compression possible).
Alternatively, suppose $\Pr[x_{i+1} | x_1,\dots,x_i] \ne 1/2$ for some $i$. Then it is possible to predict the next bit better than chance.
Those are the only two possible cases. In every case, either the next bit is predictable, or you can't compress.
This assumes you are talking about what is possible with infinite computing power (which isn't realistic, but is what is typically assumed in information theory).