This section of Scripture came to mind as I read through the thread:
2 Corinthians 4:
6 For God, who commanded the light to shine out of darkness, hath shined in our hearts, to give the light of the knowledge of the glory of God in the face of Jesus Christ.
7 But we have this treasure in earthen vessels, that the excellency of the power may be of God, and not of us.
8 We are troubled on every side, yet not distressed; we are perplexed, but not in despair;
9 Persecuted, but not forsaken; cast down, but not destroyed;
...
16 For which cause we faint not; but though our outward man perish, yet the inward man is renewed day by day.
17 For our light affliction, which is but for a moment, worketh for us a far more exceeding and eternal weight of glory;
18 While we look not at the things which are seen, but at the things which are not seen: for the things which are seen are temporal; but the things which are not seen are eternal.
I also thought about this section:
1 Peter 5:
6 Humble yourselves therefore under the mighty hand of God, that he may exalt you in due time:
7 Casting all your care upon him; for he careth for you.
8 Be sober, be vigilant; because your adversary the devil, as a roaring lion, walketh about, seeking whom he may devour:
9 Whom resist stedfast in the faith, knowing that the same afflictions are accomplished in your brethren that are in the world.
10 But the God of all grace, who hath called us unto his eternal glory by Christ Jesus, after that ye have suffered a while, make you perfect, stablish, strengthen, settle you.
11 To him be glory and dominion for ever and ever. Amen.
We are going to face afflictions, trials, tribulation in this life. God does not hide this fact from us. And in the midst of our afflictions, God is there with us, perfecting, establishing, strengthening, and settling us. He is working within us during our most trying times to help us. He never, ever leaves us alone to face the things thrown at us in this life.
So, can we have joy in the midst of our most trying times? Yes. But I also think that in those most trying of times as mentioned in the OP, we have peace and faith in our hearts as we face our difficulties.
And I wholeheartedly agree with FlSnookman7 and others who differentiate between happiness (an emotion dependent upon circumstances) and joy (fruit of the Spirit, eternal in nature).
We also have the peace that passes all understanding which guards our hearts and minds through Christ Jesus as we bring everything to God in prayer and supplication with thanksgiving.