Is every long exact sequence
$$\cdots\to\pi_{d+1}(B)\to\pi_d(F)\to\pi_d(E)\to\pi_d(B)\to\pi_{d-1}(F)\to\cdots$$
with topological spaces $F,E$ and $B$, where $F$ is a subspace of $E$ with inclusion map $i$, induced by a Serre fibration $p:E\to B$ ? By "induced" I mean that the maps in the sequence are given by 
$$p_*:\pi_d(E)\to\pi_d(B)$$
$$i_*:\pi_d(F)\to\pi_d(E)$$
and the boundary map
$$\partial:\pi_{d+1}(B)\to\pi_d(F).$$

If not, what are the conditions under which this is the case?

Thank you!