Let $X_t$ be predictable with respect to filtration $(\mathscr{F}_t:t\in[0,T] )$. If I observe the process over an interval $[0,s],0<s<T$, does that mean I can tell the value of $X_t$ over remaining period $(s,T]$?
Taking one step further, if I know $x_{0}$ does it also imply that I will know the stochastic process for all $0<t<T$?
No, both statements are in general not correct.
For example, a Brownian motion $X$ is a predictable process. But, obviously, the sample paths
$$[0,T] \ni t \mapsto X(t,\omega)$$
are not deterministically determined by the starting point $X_0=x_0$; so even if we know about the starting point, we can not predict the future of a Brownian motion. Since
$$Y_t := Y_{t+s}-Y_s, \qquad t \geq 0$$
is also a Brownian motion, we can also not expect that the information about the time interval $[0,s]$ allows us to predict the behavior in the future $(s,T]$.
Roughly speaking: If $X$ is predictable and we observe a sample path over the period $[0,t)$, then we can predict the value of $X$ at time $t$. So it's only a short glimpse into the future.
