  Hamiltonian Truncation (HT) is a numerical approach for calculating
observables in a Quantum Field Theory non-perturbatively. This approach can be
applied to theories constructed by deforming a conformal field theory with a
relevant operator of scaling dimension $\Delta$. UV divergences arise when
$\Delta$ is larger than half of the spacetime dimension $d$. These divergences
can be regulated by HT or by using a more conventional local regulator. In this
work we show that extra UV divergences appear when using HT rather than a local
regulator for $\Delta \geq d/2+1/4$, revealing a striking breakdown of
locality. Our claim is based on the analysis of conformal perturbation theory
up to fourth order. As an example we compute the Casimir energy of $d=2$
Minimal Models perturbed by operators whose dimensions take values on either
side of the threshold $d/2+1/4$.
