TITLE: Are there examples of statements that have been proven whose consistency proofs came before their proofs?
QUESTION [35 upvotes]: I'm wondering if there are examples of statements that have been proven whose consistency proofs came before the proofs of the statements themselves.
More informally, I'm wondering how promising in general is the approach of attempting a consistency proof for a statement when faced with a statement that seems true but difficult to prove.
Background:
If a statement is provable from a set of axioms, then that statement is obviously consistent (assuming the set of axioms is consistent).  So provability is stronger than consistency.  This might lead one to think that constructing a consistency proof for a statement should be strictly easier than constructing a proof.
Yet consistency proofs (at least oft-cited ones, for example those by Godel and Cohen about the Continuum Hypothesis) seem to require a high level of sophistication (though this might be a byproduct of the fact that consistency proofs like these are for the special class of statements that cannot be proven).
For statements that can be proven then, are there cases where their consistency proofs are easier or came before the proofs themselves?
Update:
Thanks a lot, everyone, for the great answers so far.  The number and existence of these examples is interesting to me, as well as the fact that they all rely on the same technique of first proving something using an additional axiom (an approach first suggested by Michael Greinecker).  That hadn't occurred to me.  I wonder if there are other approaches.

REPLY [6 votes]: Let $c(X)$ denote the cellularity of the topological space $X$, that is the supremum of the cardinalities of its families of pairwise disjoint non-empty open subsets.
In the early 60's Kurepa asked:


Is there a compact Hausdorff space $X$ such that $c(X)<c(X^2)$?


and noted that if $\mathbb{L}$ is a Suslin Line then $c(\mathbb{L})=\aleph_0<c(\mathbb{L}^2)$.
So the consistency of a positive answer to Kurepa's question has been known since the first constructions of Suslin trees using forcing by Jech and Tennenbaum in the late 60s. Later Galvin was able to get a compact CCC space whose square is not CCC from the Continuum Hypothesis.
And while the existence of a CCC space whose square is not CCC is independent from ZFC (an arbitrary product of CCC spaces is CCC under $MA_{\omega_1}$ for example), in the mid 80s Todorcevic surprised everyone by constructing a compact space $X$ such that $c(X)<c(X^2)$ in ZFC.
http://archive.numdam.org/article/CM_1986__57_3_357_0.pdf
Todorcevic's paper contained many other ZFC constructions of objects whose consistency had been previously known, including higher cardinal generalizations of $S$-spaces and $L$-spaces.