I am not aware of much evidence for arbitrary high rank elliptic curves. Silverman in his book (Arithmetic of elliptic curves) gives as evidence the lack of evidence for the opposite statement. I guess that the main evidence is the fact that every now and then someone comes up with a new record.
The function field case is some sort of evidence, however, most constructions rely on supersingular elliptic surfaces or supersingular Fermat surfaces. These objects are quite far away from characteristic zero objects. (One exception to this is the construction by Bouw-Diem-Scholten.) Shioda tried to use Fermat surfaces in order to produce high rank elliptic curves over $K(t)$, with $char(K)=0$$\operatorname{char}(K)=0$, but the best you can get from that is rank 68 if $K$ is algebraically closed, for $K=\mathbb{Q}$ this maximum is much lower.
A second piece of evidence is the fact that we can construct big Selmer groups. We can find arbitrarily many elements of order m in the m-Selmer group, for $m=2,\dots,10,12,13,16,25$. However, every construction of large Selmer groups seems to allow a refinement that makes the Tate-Shafarevich group large. (For some reason nobody wrote this up for the composite $m$ mentioned above, but I am quite sure that this has been done.) Matsuno gave a construction s.t. for each prime number $p$ and each field $K$ containing a degree $p$ Galois extension of $\mathbb{Q}$ one can produce a series of elliptic curves $E/\mathbb{Q}$ the group $S^p(E/K)$ can be arbitrarily large, but also this construction allows a refinement to get big Tate-Shafarevich groups.
