There's plenty of literature on results that say that interpolation errors are small for smooth functions. But, I can't find much information about the converse, that is - does convergence of interpolant imply that the function is regular?
Here's some examples of the theorems -
Theorem 1 - Global approximation
For an integer $\nu \ge 0$, let $f : [-1,1] \to \mathbb{R}$ and its derivatives through $f^{(\nu-1)}$ be absolutely continuous on $[-1,1]$ and suppose the $\nu^\text{th}$ derivative $f^{(\nu)}$ is of bounded variation $V$. Then, for $n> \nu$, its Chebyshev interpolants $\{p_n\}$ satisfy $$\| f - p_n\|_\infty \le \frac{2V}{\pi \nu(n-\nu)^\nu}. $$ Here, Chebyshev interpolant $p_n$ of a function $f$ is a polynomial of degree $n$ which agrees with $f$ on $n$ Chebyshev points.
Source - Page 64, Approximation theory and approximation practice - Nick Trefethen
In this case, the converse question would be - if the Chebyshev interpolants converge with the above rate for $n > \nu$, is $f^{(\nu-1)}$ absolutely continuous and $f^\nu$ of bounded variation?
Theorem 2 - Finite Elements/splines
Theorems of this kind are often written as
Suppose $\mathcal{T}_h$ is a regular family of triangulations of $\Omega$ which is a convex polygonal domain, then the finite element approximation $u_h \in X_h^k (k \ge 1)$ of $u \in H^{k+1}(\Omega) \cap H_0^1(\Omega)$, satisfies $$\|u - u_h\|_{H^1(\Omega)} \le C h^k |u|_{k+1,\Omega} $$
Here's a self-contained, 1-D version
Let $I = [a,b]$ be an interval. Write $I$ as a union of closed intervals $\{I_j\}$ which can intersect only at the end points. Define reference maps $T_h^j : I_j \to [0,1]$ which are linear and bijective. Let $n \ge 0$ and let $\{x_k\}_{k:0}^n$ be $n+1$ equispaced points in $[0,1]$. Define an interpolation operator $\mathcal{I}_h^n : H^1(I) \cap C(I) \to H^1(I)$ so that, for any $f \in C(I)$, we have $\mathcal{I}_h^n f |_{I_j} \in \mathbb{P}_n$ and the local polynomial agrees with $f$ at equispaced points in $I_j$, i.e., $$\mathcal{I}_h^nf(T_h^j(x_k)) = \mathcal{I}_h^n, \qquad \forall k,j.$$ Then, if $f \in H^{n+1}(I)$, we have $C$ independent of $h$ such that $$\|f-\mathcal{I}_h^nf\|_{H^1(I)} \le C h^n |u|_{n+1(\Omega)}$$ where $|\cdot|_{n+1}$ is the semi-norm corresponding to $(n+1)^{th}$ derivative.
In this case, the converse would again be - If the interpolation error is of order $n+1$ for an $H^1(I)$ function, is the function in $H^{n+1}$(I)?
This is partly motivated due to the theorem in Fourier series which does have a converse
Theorem 3
Let $f \in L^2([0,2\pi])$ be periodic, let $\hat{f_k}$ be the $k^\text{th}$ Fourier coefficient of $f$. Then, $f \in H^m([0,2\pi])$ if and only if $(1+|k|^2)^{1/2} \hat{f_k} \in \ell^2(\mathbb{Z})$.
I highly doubt your proposals can hold. The functions you interpolate with (eg. Chebyshev Polynomials) look only at values of a function at sampled points. You can have a family of functions that coincide on some N points but are for example not uniformly bounded. Chebyshev polynomial would be the same for every one of those functions. Only by assuming regularity conditions of the family of functions we interpolate and the length of the interval, we can conclude some interpolation error bounds.
As for the Fourier series case, note that every coefficient is obtained from a global information about the function. Because of this, we can say something about its regularity on the whole interval.