Yes, although this isn’t a rock-paper-scissors situation. Just because tactics were developed to allow infantry troops to defend against horse archery (and cataphract) tactics to some extent does not, of course mean that these units lost all effectiveness on the battlefield or that armies using the counter-tactics were guaranteed success. The ultimately continuing effectiveness of cavalry, of both the shock and archery varieties, is shown by the fact that the Romans in late antiquity and in the medieval (Eastern Roman) era gave horse archery and heavy shock cavalry a prominent role in their own armies.
However, it turns out that well-balanced infantry armies that used various differently armed soldiers to support one another were a lot better suited to resisting cavalry tactics than were armies overprovided with legionaries and underprovided with other supporting troops. In more balanced armies, the legionaries could protect the supporting troops from heavy shock cavalry and the supporting troops could stop horse archers from endlessly bombarding the legionaries without being threatened in reply.
Clever use of tactics and terrain also could get the advantage for Roman infantry. Ventidius inflicted several serious defeats on Parthian cavalry largely by exploiting hilly terrain (and shadowing cavalry-heavy armies from the hills had been a Roman fallback tactic for ages, at least since they were fighting Hannibal in Italy).
The basic anti-cavalry tactic used catastrophically by Crassus (probably due to lack of enough of the right kind of non-legionary troops, and due to letting those he did have get isolated from the rest of his army) and with better results by, e.g., Antony and Corbulo – defensive squares in which heavy infantry protects light missile infantry from melee and the light troops protect the heavy infantry from ranged attack, while friendly cavalry can sally in and out – remained an important element of Roman anti-cavalry tactics for many centuries. It can still be found prescribed in the Praecepta Militaria, written for Eastern Roman warfare against Arabs a thousand years after Carrhae.
It's almost unfair to assume that Rome was ever seriously threatened by steppe nomads on the basis of their cavalry armies. This may have come from the disaster of Carrhae in 53BC by the Parthians, as outlined by /u/Aithiopika. The supposed superiority of the Parthian cavalry was negated in the retaliatory campaign by Mark Antony, with Ventidius leading the initial advance through Anatolia in 39BC. At the Battle of the Cilician Gates, Ventidius positioned his legionaries, supported by slingers and javelin-throwers, uphill. The Parthian cavalry did not wait for reinforcements and charged uphill. Their horse archers were ineffective against the legionary testudos. The Romans returned with their own volleys and charged downhill, routing the lightly-armoured horsemen. This was followed up at Amanus Pass, where Ventidius used his cavalry units to lure the Parthians into a bottleneck, where they were surrounded and wiped out.
While the Parthian Wars ultimately ended in a stalemate, the Romans clearly proved that they were capable of fighting cavalry armies when they had the proper balance of troops and good leadership and positioning.
Fast forward to the Battle of the Catalaunian Plains in 451AD, and we see that the Eastern Roman general Aetius, who had an intimate knowledge of Hunnic warfare as a hostage and later a commander of Hunnic mercenaries, was able to negate the Hunnic horse archers by keeping his men in a shield wall while missile troops inflicted casualties on the light horse archers as they rode past. Combined with later events, this ultimately ended in a victory for the Romans against the Huns.
The Byzantines would make use of their own horse archers and shock cavalry in their armies. pretty much putting them on equal footing against similar threats.