It’s not that simple. The Goths had a long, and complicated, history with the Romans even before the Huns showed up. During the crisis of the third century the Goths led several large scale raids into Roman territory. Them the Goths made a treaty with Constantine after he defeated them in battle. Emperor Valens however went to war with the Goths in 367 because they allegedly violated said treaty by supporting the revolt of Procopius, who was a cousin to the deceased emperor Julian. In 369 he crossed the Danube and devastated the country side forcing the Gothic king Athanaric to meet him in battle. The Romans were victorious and the Goths had to sign another treaty. This treaty, however, was very unfavorable for the Goths and probably played some role in destabilizing the region.

The Goths then began to have internal conflict between Christians and pagans. It is possible that some of the Roman soldiers stationed on the Danube supported the Christian faction in some way shape or form. This only served to further weaken the Goths.

It should also be noted that there was two main gothic tribes at this point in time. The Thervingi and the Greuthungi. Because the Greuthungi lived further east they encountered the Huns first, and were soundly defeated. The Thervingi then decided they would be better off moving en masse to the Danube and asking Emperor Valens got permission to settle within the Empire.

Valens was also in the middle of a war with the Sassanids, and was in need of more men. The Goths, now willingly presenting themselves to him, had at least 10,000 warriors with them. Thus Valens readily have them permission to settle, and This is where things really get complicated. Valens ordered the Roman officials overseeing the river crossing to take all of the weapons that the Goths had brought with them. For whatever reason, however, they failed to do this. The Goths were also treated poorly, and according to Ammianus may have even been forced to sell their own children into slavery just to be able to afford food. He is quite explicit in his blame of these officials for what was too follow (as mentioned in nightsky555’s comment).

“Their treacherous greed was the source of all our evils. I say nothing of other crimes which these two men, or at least others with their permission, with the worst of motives committed against the foreign new-comers, who were as yet blameless; but one melancholy and unheard-of act shall be mentioned, of which, even if they were their own judges of their own case, they could not be acquitted by any excuse. When the barbarians after their crossing were harassed by lack of food, those most hateful generals devised a disgraceful traffic; they exchanged every dog that their insatiability could gather from far and wide for one slave each, and among these were carried off also sons of the chieftains. "

Ammianus Marcellinus, Res gestae XXXI 4.10-11

The main Roman in charge, Lupicinus, then decided the Goths should be moved to Marcianople. The problem with this however was that it required the majority of the Danube garrison to do so. This left the river largely undefended, and would allow the Greuthungi, who had previously been denied by the Emperor, to cross. When the Thervingi reached Marcianople Lupicinus invited their two leaders, Fritigern and Alavivus, to a feast. This was a ruse, however, and he actually tried to kill them both. While this was happening the Thervingi outside the city gates began to grow impatient and demanded to see their leaders. According to Ammianus, Fritigern made a deal with Lupicinus and promised to calm down his people should he be allowed to leave unharmed, it’s possible that he also allowed the Romans to take Alavivus. The plan either did not work, or Fritigern had lied about his intentions from the get go. Either way the Goths slaughtered the Roman soldiers at Marcianople.

Sometime in 377 some of Valens general met the Goths in battle near the Haemus Mountains, but this battle was inconclusive. The Greuthungi Goths had already joined with the Thervingi after Marcianople, and now they were joined by some Alans as well. This allowed them to get out of the mountains and move towards Adrianople. Around this same time Valens made peace with Persia so he could focus on the Goths.

In 378 Valens began to march his army, who were now veterans of the Sassanian war, towards Adrianople. He had initially planned to wait for his cousin, the Emperor Gratian, to show up with western reinforcements. Ammianus, however, tells us that Valens has begun to grow jealous of his own generals, as well as his cousin (he was delayed because he had to fight off an invasion by the Alemanni), because he did not yet have his own great victory. According to Zosimus he was persuaded to engage the Goths before Gratian could arrive, because his generals, who apparently did not get along with each other, convinced him to do so. Valens’ scouts had also grossly underestimated the total strength of the Goths.

The Romans were of course defeated and now had a permanent foothold in the Empire. Valens successor Theodosius I, also fought a war against the Goths. While he did win the war, his victory wasn’t decisive enough to force the Goths out of the Empire. As such they were given land to settle in Thrace. They were even exempted from taxes, and allowed to govern themselves, so long as they provided troops to the emperor.

This of course only lasted for about 10 or so years before the Goths rebelled again. To pacifist them their leader Alaric was appointed Magister Militum of Illyria. It’s speculated that one of the reasons Alaric rebelled against Emperor Arcadius, is because he was going to lose his title of Magister Millitum. This title is what allowed him to pay and feed his men, so he decided to invade Italy. He invaded in 401, but was defeated by Stilicho twice in the year 402. But the Roman senate still had to pay the Goths a substantial tribute. Alaric and his Goths may have reentered Roman service after this. In 406 another Gothic “king”, Radagaisus, invaded Italy. He was also defeated by Stilicho. In 408 the western emperor Honorius had Stilicho executed, and then he incited a slaughter of the families of his gothic troops (many of whom had been in Radagaisus’ invading army). The men whose families were killed then joined with Alaric.

Alaric then marched on Rome, and laid siege to the city. The senate was once again forced to pay him tribute. Honorius, however, refused to give in to Alaric’s demands, and so Alaric besieged Rome again in 409. In 410 he besieged the city for a third time and this time actually took the city. Alaric died a few months after this, and the Goths marched back up into Gaul where they would remain for several decades as a somewhat autonomous state. Then, of course, by 493 the Visigoths controlled most of Spain, and the Ostrogoths Italy.

In short it’s not as simple as “the Huns attacked them, and then the Goths continually betrayed the Roman Empire.” The Goths sought refuge within the Empire’s borders, but were subjected to poor treatment and even worse conditions. Their leaders were almost assassinated, and they weren’t given much food. When enough was enough they rebelled.

Sources: Peter Heather, The Fall of the Roman Empire: a New History of Rome and the Barbarians.

Peter Heather, Empires and Barbarians: Migration, Development and the Birth of Europe

Guy Halsall, Barbarian Migrations and the Roman West, 376-568

Ammianus Marcellinus, The Later Roman Empire