If a new hash function were made that simply truncated sha-256 to the first N bits, how much insight would it provide into sha-256?
If this provided any insight, we would consider SHA-256 to be broken.  In general, we expect the $n$-bit truncation of SHA-256 to resemble an $n$-bit uniform random function.  For example, finding a 10-bit partial preimage costs an expected ${\approx}2^{10}$ trials; finding a 10-bit partial collision costs an expected ${\approx}2^5$ trials.

That said, there are often simplified versions of cryptographic primitives for study:


One obvious simplification of SHA-256 is to reduce the number of rounds from 64—for SHA-256 as for most cryptographic primitives with rounds, the first (and, for unbroken cryptosystems, only) attacks published in the literature are on reduced-round variants.
Aside from varying the number of rounds, Keccak supports any power-of-two word size from 1-bit to 64-bits; SHA-3 is based on 64-bit Keccak with a 1600-bit state, but you could study 1-bit Keccak with a 25-bit state (with any number of rounds).
The SHA-3 candidate BLAKE came with several toy versions BLOKE, FLAKE, BLAZE, and BRAKE.