Experimental exactness of Schrödinger equation for more than 100 particles

Despite it being impossible to exactly simulate a large number of particles on a computer, physicists have found many tricks to approximately simulate it. From Quantum Monte Carlo to Density functional theory to the whole field of Tensor networks and many others, there is a plethora of methods to obtain accurate predictions for the physics of many particles that doesn't involve attacking the problem directly (which would be impossible).

As an example, a single particle is described by a $d$ dimensional vector space, so an $N$ particles state is described by an unmanageable $d^N$ dimensional vector. It turns out though that physically relevant states of $N$ particles (i.e. ground states of local Hamiltonians) have some nice properties that allows them to be tractable. Specifically, they have relatively low entanglement (i.e. since the physics is described by interaction between nearby particles, far away particles will not be very correlated). This allows to get an accurate description of such a state with a number of parameters that scales polynomially instead of exponentially in the system size. This is only possible because the physically relevant case is not the general case. This observation is where the techinques associated with tensor networks and matrix product states stem from.

If n is enough big then there is no computer which can give a numerical approximation of the solution to the corresponding shrodinger equation.

Not true. In nuclear physics we routinely get good numerical approximations to the properties of nuclei that have 100 particles. In condensed matter physics, it's pretty routine to get good models of $10^{23}$ particles.

There may be some sense in which this claim is true, but if so, the burden is on you or the person making this claim to define what they really mean.

Statements such as

Quantum mechanics is the most exact physical theory

are usually based on precision tests of quantum electrodynamics, where QED predicts values for fundamental parameters such as the fine-structure constant that agree with experimentally measured values to a very high degree of precision - typically a few parts in a billion.