Can a simulation be "a little bit" nonlinear?

Not in the solver's eyes — a given analysis either uses a linear solver (one matrix, one solve) or a nonlinear solver (incremental, iterative). What is graded is how much the nonlinear behavior actually matters: a slightly-too-large deflection assumption in an otherwise stiff, small-deflection problem might introduce only a negligible error if run as linear, while the same assumption on a thin, flexible part could be badly wrong. The rule of thumb many analysts use is checking deflection against a fraction of the part's smallest relevant dimension (often cited around 1/20 to 1/10 of thickness or span) as a first-pass linearity screen.

Does nonlinear FEA always give a "more correct" answer than linear FEA?

Only when the underlying physics is actually nonlinear. If a structure genuinely stays in small deflections, stays below yield, and has no contact status changes, a linear solve gives the mathematically exact same answer a nonlinear solve would converge to — just far faster, because there is nothing for the extra iterations to correct.

What is the difference between geometric nonlinearity and buckling analysis?

Buckling analysis (linear eigenvalue buckling) predicts the load at which a structure would theoretically lose stability and the associated mode shape, using a linearized approximation — it does not capture post-buckling behavior. A full geometrically nonlinear (large-deflection) analysis, sometimes combined with an artificial imperfection, can trace the actual load-deflection path through and past that instability, including snap-through and post-buckling load redistribution, which linear eigenvalue buckling cannot show.

Why does a nonlinear solve sometimes fail to converge?

Non-convergence usually means Newton-Raphson iterations at some load increment can't drive the residual force below tolerance — common causes include too large a load increment for a highly nonlinear response (rapid contact status changes, near-buckling stiffness loss), poorly conditioned contact definitions (initial penetration, mismatched friction settings), or a genuinely unstable equilibrium path (structural collapse) that the solver is correctly failing to find a stable solution for.

Is a "nonlinear material" the same thing as a "plastic material"?

No — plasticity (permanent, path-dependent deformation past yield) is one specific type of material nonlinearity, but not the only one. Hyperelastic materials (rubber, elastomers), viscoelastic materials (creep, stress relaxation), and materials with a nonlinear-but-still-fully-elastic stress-strain curve are all materially nonlinear without ever undergoing plastic (permanent) deformation.

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Linear vs. Nonlinear FEA

It isn't about whether your material is elastic. It's about whether the stiffness matrix stays the same number from the first load increment to the last.

A common assumption is: "my part is steel, staying well under yield, so my FEA is linear." Material behavior is only one of three things that can break the linear assumption. A thin steel bracket deflecting several times its own thickness, two elastic parts that start apart and come into contact under load, a bolt with pretension — none of these involve yielding, and all three require a nonlinear solve. The real dividing line is narrower and more mechanical than "is the material elastic": does the stiffness relating load to displacement, [K], stay constant throughout the load, or does it change as the structure deforms?