Suppose there are two directed systems $(V_i,\phi_i:V_i\rightarrow V_{i+1})$ and $(W_i,\psi_i:W_i\rightarrow W_{i+1})$ of vector spaces with $i \ge 1$. Assume that the direct limits up to term $n$ are isomorphic, ie. $\lim_{\phi_i, i \le n} V_i$, $\lim_{\psi_i, i \le n} W_i$ are isomorphic. Is it true the entire direct limits $\lim_{\phi_i} V_i, \lim_{\psi_i} W_i$ are isomorphic? If this is false, can some weak conditions we added to make it true?
Maybe an easier case is when $V_i = W_i$ and $\psi_i = \phi_i \circ f_i$ for some automorphism $f_i$ of $V_i$. See my past question: Direct limit isomorphism. Another possible case is when $V_i$ are finite-dimensional vector spaces or $V_i$ are finitely-generated abelian groups.
(I assume you mean to say "for all $n$" somewhere in that first paragraph)
The truncations are rather trivial; the diagram
$$0 \to 1 \to \ldots \to n$$
has a final object; consequently, for any functor $F$ from this diagram to any category
$$\operatorname{colim}_{i \leq n} F(i) \cong F(n) $$
So, your condition that the truncated colimits be equal is simply that the two systems are termwise isomorphic.
In the category of abelian groups
$$\operatorname{colim}\left( \mathbb{Z} \to \frac{1}{2}\mathbb{Z} \to \frac{1}{4} \mathbb{Z} \to \ldots \right) \cong \mathbb{Z}\left[1/2\right] $$
where $a \mathbb{Z}$ means the group of integer multiples of $a$, and $\mathbb{Z}[1/2]$ means the (additive group of) the ring generated $1/2$. That is, the group of rational numbers whose denominator is a power of $2$.
The maps are the inclusions.
Also,
$$\operatorname{colim}\left( \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z} \to \ldots \right) \cong \mathbb{Z} $$
where the maps again are inclusions.
These two colimits are clearly not isomorphic, despite the diagrams being termwise isomorphic!
There is a very simple condition to ensure that the colimits are isomorphic — rather than ask for the terms to be isomorphic, ask for them to be naturally isomorphic.
A morphism between the two systems is a family of maps $f_i : V_i \to W_i$ with the property that $$ \psi_i \circ f_i = f_{i+1} \circ \phi_i $$ (that is, a natural transformation between the corresponding functors)
Any morphism between systems induces a corresponding morphism
$$ f_* : \left(\operatorname{colim}_i V_i\right) \to\left(\operatorname{colim}_i W_i\right) $$
When every $f_i$ is an isomorphism, then $f_*$ is also an isomorphism.