Does Ezekiel 18:20, which says that “a son himself will bear nothing because of the error of the father,” contradict Exodus 20:5, which says that Jehovah brings “punishment for the error of fathers upon sons”?

 
There is no contradiction. One statement focuses on the individual’s accountability, and the other acknowledges the reality that a person’s error may have consequences affecting his descendants.
 
The context of Ezekiel chapter 18 shows that personal accountability is being stressed. “The soul that is sinning—it itself will die,” states verse 4. What about a man who is “righteous and he has executed justice and righteousness”? “He will positively keep living.” (Ezek. 18:5, 9) Thus, after reaching an age of accountability, each individual is judged “according to his ways.”—Ezek. 18:30.
 
This principle is illustrated in the case of a Levite named Korah. During Israel’s wilderness journey, Korah became dissatisfied with his service privileges. In an effort to secure priestly duties for himself, Korah and some others rebelled against Jehovah’s representatives, Moses and Aaron. For presumptuously reaching out for this office—a privilege they were not entitled to—Jehovah put Korah and his rebel forces to death. (Num. 16:8-11, 31-33) Korah’s sons, however, did not join in the rebellion. God did not hold them accountable for their father’s sin. Their loyalty to Jehovah resulted in their own lives being spared.—Num. 26:10, 11.
 
But what about the warning at Exodus 20:5, part of the Ten Commandments? Again, consider the context. Jehovah inaugurated the Law covenant with the nation of Israel. After hearing the terms of the covenant, the Israelites declared publicly: “All that Jehovah has spoken we are willing to do.” (Ex. 19:5-8) The nation as a whole thus entered into a special relationship with Jehovah. So the words at Exodus 20:5 were fundamentally directed to the whole nation.
 
When the Israelites remained faithful to Jehovah, the nation benefited and enjoyed many blessings. (Lev. 26:3-8) The opposite also held true. When the nation of Israel rejected Jehovah and went after false gods, he withdrew his blessing and protection; the nation suffered calamity. (Judg. 2:11-18) Granted, there were some who maintained their integrity and kept God’s commandments despite the nation’s idolatrous course. (1 Ki. 19:14, 18) Faithful ones likely experienced some hardship because of the nation’s sins, but Jehovah expressed loving-kindness toward them.
 
When Israel became so flagrant in violating Jehovah’s principles that his name became an object of derision among the nations, Jehovah determined to punish his people by allowing them to be taken captive to Babylon. Of course, this included punishment of individuals and of his people as a group. (Jer. 52:3-11, 27) Indeed, the Bible shows that Israel’s collective guilt was so great that three, four, or possibly more generations were affected by the misdeeds of their forefathers, just as Exodus 20:5 states.
 
God’s Word also contains accounts where individual families were affected by parental misconduct. High Priest Eli offended Jehovah by allowing his “good-for-nothing,” immoral sons to remain as priests. (1 Sam. 2:12-16, 22-25) Because Eli honored his sons more than Jehovah, God decreed that Eli’s family would be cut off from the high priesthood, which happened starting with his great-great-grandson, Abiathar. (1 Sam. 2:29-36; 1 Ki. 2:27) 

The principle of Exodus 20:5 is illustrated by the example of Gehazi as well. He abused his position as Elisha’s attendant in order to benefit materially from the curing of Syrian General Naaman. 

Through Elisha, Jehovah pronounced judgment, saying: “The leprosy of Naaman will stick to you and your offspring to time indefinite.” (2 Ki. 5:20-27) So Gehazi’s descendants were affected by the consequences of his wrongdoing.
 
As Creator and Life-Giver, Jehovah has every right to determine what punishment is just and appropriate. The above instances show that children or descendants may feel the ill effects of their ancestors’ sin. However, Jehovah “hears the outcry of the afflicted ones,” and individuals who earnestly turn to him may receive his favor and even some measure of relief.—Job 34:28.