Probably the closest thing to what you're looking for is Chaitin's proof of the incompleteness theorem, which shows that for any formal system $S$, there is a constant $L$ such that the statement "$K(s) > L$" is unprovable in $S$ for all strings $s$ (here $K$ denotes Kolmogorov complexity). The vast majority of such statements are true "at random" because a random string will have high Kolmogorov complexity.
However, you didn't ask whether there exists a family of statements that can be regarded as being "true by accident"; you asked whether a specific statement (the graph reconstruction conjecture) can be regarded as being "true by accident." So Chaitin's incompleteness theorem doesn't quite address your question as stated.
It is certainly possible, in some situations, to construct a heuristic probabilistic model that predicts that certain things ought to be true just for "random reasons." Perhaps the most famous example is Cramér's random model for the primes, which can be used to give heuristic "proofs" of various number-theoretic conjectures; e.g., one can use the model to predict that there will be only finitely many primes with such-and-such a property, because the probability that a prime $p$ has the property decreases rapidly to zero as $p\to\infty$. If a conjecture is predicted by such a model, and also "happens" to be unprovable in your favorite axiomatic system for mathematics, then it might be tempting to say that it is "true by accident."
This kind of thing could happen, but we don't know of any good examples involving "naturally occurring" mathematical conjectures. So it's all pure speculation at this point. Certainly we have no objective evidence that any particular mathematical conjecture isn't worth working on for this reason. I think it would be interesting, though, if you could develop a heuristic probabilistic model for graph theory, in the spirit of Cramér's model, that could "predict" various well-known graph-theoretic conjectures.