Why does God allow bad things to happen to good people?

First the question is flawed, it assumes that according to God there are “good people” in the same sense we think of people as “being good”. This is not the case, Romans 3:23 “For all have sinned and fall short of the glory of God”. So in God’s eyes we all fall short of “good people”, we are sinful.

So why if he is a “loving God” does he allow bad things to happen if he is all powerful and could stop them?

It is a good question but again it is a flawed question. He is all powerful and all just. He has given man a choice, and allowed the consequences of that choice. In Genesis 1 God created a perfect world, one that was designed to allow the best for human kind. In Genesis 3 human kind, Adam and Eve, make a choice that effects human kind forever. They sin against a Just God, and being Just he must punish the sin.

Then to Adam He said, “Because you have heeded the voice of your wife, and have eaten from the tree of which I commanded you, saying, ‘You shall not eat of it’:
“Cursed is the ground for your sake;
In toil you shall eat of it
All the days of your life.
Genesis 3:17

This continues today as stated in Romans
19 For the earnest expectation of the creation eagerly waits for the revealing of the sons of God. 20 For the creation was subjected to futility, not willingly, but because of Him who subjected it in hope; 21 because the creation itself also will be delivered from the bondage of corruption into the glorious liberty of the children of God. 22 For we know that the whole creation groans and labors with birth pangs together until now.
Romans 8:19-22 (NKJV)

The result of the original sin has a consequence in the world we live in today. We still live in a world that people can look at and see God’s wonders and be without excuse, but it is a fallen world.
Rom 1:20
20 For since the creation of the world His invisible attributes are clearly seen, being understood by the things that are made, even His eternal power and Godhead, so that they are without excuse, 21 because, although they knew God, they did not glorify Him as God, nor were thankful, but became futile in their thoughts, and their foolish hearts were darkened.
Romans 1:20-21 (NKJV)
The heavens declare the glory of God; And the firmament shows His handiwork.
Psalm 19:1 (NKJV)

The current world we live in suffers from the curse, from the sin of man, and we suffer with it at times. We can subdue it to a certain point but ultimately it is a fallen world.

Don’t be dismayed though! There is hope! Paul states in Romans 8:18 “For I consider that the sufferings of this present time are not worthy to be compared with the glory which shall be revealed in us.” There is a hope of the future, as Answers In Genesis puts it the 7th “C” Consummation.

“Those who have repented and placed their faith in the finished work of Christ look forward to the consummation of all things in the future, when the Curse will be removed and God will provide the new heavens and the new earth for His children. There will be no death or mourning or crying or pain (Revelation 21:4).” (AIG Website)

So what must we do?
Therefore, my beloved brethren, be steadfast, immovable, always abounding in the work of the Lord, knowing that your labor is not in vain in the Lord.
1 Corinthians 15:58 (NKJV)

As Paul did, we do our best to live for Christ and wait on the hope of a New Heaven and a New Earth.

Hold Steadfast!