I had a court case against a company and the court forced the company to pay me some money plus an interest on that money. Can I take this interest knowing that the court order is less than my right because I did not have enough papers proving my case and because of a mistake made by my lawyer?

Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
You are permitted to take from the company what is your actual right without adding to your right any interest or income. Taking something more than your right against a period is Riba (usury/interest). The court decision does not make it Halal. The Prophet said: "If I decide in favour of a person not entitled to the thing, I do not allow him but a piece of fire".
If the company does not paid you all your rights and you could not prove your right in court for any reason such as lack of legal papers or for the weakness of advocate at that time you may take from interest your right and your right only. This is the opinion of the majority of the Muslims scholars. The evidence for this opinion is the saying of Allah - Allah Says (interpretation of meaning): {The recompense for an evil is an evil like thereof, but whoever forgives and makes reconciliation, his reward is due from Allâh. Verily, He likes not the Zâlimûn (oppressors, polytheists, and wrong-doers, etc.).}[42: 40].
He also Says: {And if you punish (your enemy, O you believers in the Oneness of Allâh), then punish them with the like of that with which you were afflicted. But if you endure patiently, verily, it is better for As-Sâbirin (the patient ones, etc.)}[16: 126]
He also Says: {.And indeed whosoever takes revenge after he has suffered wrong, for such there is no way (of blame) against them.}[42: 41].
Some scholars say that one should take back his right from the person who had taken some thing from him unjustly or robbed, or the one who delays payments deliberately. The evidence of this opinion is the Hadith that the Prophet said: "Whoever among you sees an evil, he must change it with his hand, if he is unable to do so, then with his tongue, and if he is unable to do so, then with his heart". No doubt, whoever able to stop unjustice and able to get back the rights of the oppressed person and does not do so, he commits sin since he disobeys the Prophet (Peace and blessing be upon him).

Allah knows best.