Does the Bible say that if you have broken one commandment you have broken them all?

Yes. James 2:10-11 says:

For whoever keeps the whole law but fails in one point has become accountable for all of it. {11} For he who said, “Do not commit adultery,” also said, “Do not murder.” If you do not commit adultery but do murder, you have become a transgressor of the law.

Indeed, if we do not love our neighbour as ourself (James 2:8-9) then we are breaking the whole law. John puts it like this:

Everyone who makes a practice of sinning also practices lawlessness; sin is lawlessness. (1John 3:4)

Even though Christians don’t need to keep the Law any more (Rom. 10:4), if we sin we are law breakers. This is to encourage us to try to live a life like God’s: a life away from sin; a life of love towards others, because God is love (1John 4:8,16).

Thankfully, when we  sin God is willing to forgive us if we repent (1John 1:9). Part of the purpose of the Law was to help us realise that they couldn’t be righteous without faith in God’s grace to forgive (Gal.3:19-24); we are reminded of this when we fail to love our neighbour as ourselves,  and we can continue to rely on God’s grace to forgive us.