Consider first the usual Fock space (countably $\infty$-dim vector space) whose basis states $|n\rangle$ are labeled by nonnegative integers $n\geq 0$. This space admits what are called minimal projections: matrices $|n\rangle\langle n|$ which serve as the fundamental building blocks of all matrices that are diagonal in this basis. These building blocks are trace one, and cannot be further divided into sums of two other projections which have lower trace. The presence of such minimal projections implies that any von Neumann algebra on this space is of Type I.
Now consider a tensor product space of $n$ qubits (the vector space $\mathbb{C}^2$) with $n$ being arbitrarily large, and with allowed operators acting nontrivially only on a finite subset of the qubits. Such a space does not admit a minimal projection. Consider a projection
$$
\Pi = |0\rangle\langle0|^{\otimes k}\otimes I^{\otimes (n-k)}
$$
onto the basis state $|0\rangle\langle 0|$ for the first $k$ qubits and identity $I$ on the remaining $n-k$. The trace of this projection is $2^{n-k}$. It is not minimal because we can resolve the identity on the $k+1^{\text{st}}$ qubit and construct projections (with $\ell\in\{0,1\}$)
$$
\Pi_{\ell} = |0\rangle\langle0|^{\otimes k}\otimes |\ell\rangle\langle\ell|\otimes|0\rangle\langle0|^{\otimes (n-k-1)}
$$
that have smaller trace and sum up to $\Pi$. The lack of a minimal projection makes this space a candidate for housing Type II factors.
This ability to always resolve identities of other qubits allows us to keep cutting up projections into finer and finer pieces. This difference between the two spaces reminds me of the fact that any positive real number can always be expressed as a sum of two other positive reals, while there exists an element of the integers (namely, $1$) that cannot be expressed as a sum of two positive integers.
