Q: A conjecture in which both "if" and "only if" are near misses [Migrated from Math Stack Exchange]
More than a year ago, I posted the following on the Math Stack Exchange.  

Consider $2^n-1$. Based on checking a few small numbers for $n$ (in
  fact, the first ten natural numbers) we might "conclude" that "$n$ is
  prime if and only if $2^n-1$ is prime." However, further investigation
  shows that for $n=11$, $2^n-1$ is not prime. Thus our claim that "if
  $n$ is prime then $2^n-1$ is prime" is a near miss. However, we can
  prove its reverse, that "if $2^n-1$ is prime then $n$ is prime".   I
  am looking for some elementary number theoretic examples that both
  directions of our eventually fake biconditional are near misses.

To my great surprise, I haven't received even a near miss comment, let alone an answer. I know that the notion of "near miss" is disputable. Thus you might want to ignore my example above, and give an example of a biconditional statement  (number-theoretic or not, historical or not, made-up or not) that seems to be true for some reasons (e.g., observing a number of cases, or intuition, etc) in both directions, but then it turns out that it is, in fact, false in both directions.    

A: False claim: A Hausdorff topological space is compact if and only if it is sequentially compact.
It's believable if your intuition of Hausdorff spaces comes entirely from metric spaces (where the claim is, in fact, true). However, both directions are false for different reasons:
Counterexample I: the Alexandroff line is sequentially compact but not compact.
Counterexample II: the (Tychonoff) product of $|\mathbb R| $-many copies of the unit interval is compact but not sequentially compact.
