"God doesn't always heal."
I think these words are meant to convey that Christians should seek the will of God, as we don't know if he's allowing illness for some purpose that's ultimately for the believer's good. Or perhaps these words are meant to remind us that we aren't entitled to blessings or in any position to demand them from God.
A similar, well-intentioned statement is: "Our greatest spiritual growth comes through trials." This is probably meant to remind us that trials cause us to run to God and may reveal a complacent heart.
Indeed, these are important lessons for the believer to reflect on when life is going well. But such statements may be deeply harmful when made to those who are struggling through an illness or difficult life circumstances.
That's because even strong Christians may be tempted to doubt God's love at times when they're overwhelmed by physical or emotional pain. Hearing "God doesn't always heal" might cause a person to focus on the idea that she's the exception—and then question, Why doesn't God want to heal me? Am I less important to God, or less worthy than others?
In this way, these statements may paint an ugly picture of God as preferential, or indifferent, or harsh—like a drill sergeant who uses only extreme methods to whip us into shape, and who favors the strong.
There's a truer statement to be made that reflects God's love for us: God always heals. Whether it's through temporary healing provided in this present life, or in the complete physical and emotional healing that he will provide in eternity (Revelation 21:4), God heals all wounds.1