It was, though usually the availability of multiple sources prevented the city from feeling the pinch. By the late 40's BCE (I'm not sure exactly when the episode you watched is set - it's been a long time since I saw the show), the chief sources of supply were Sicily, North Africa, and Sardinia. Adequate grain production was not typically a problem; although one or two or these areas might suffer substandard harvest in a given year, the variegated climate of the Mediterranean ensured that there was almost always a bumper crop somewhere. The problem lay in getting all that grain to Rome.
Between 43 and 36 BCE, Sextus Pompeius (who was, as you know from the show, a bitter opponent of Octavian and Antony) controlled Sicily and the surrounding seas. This position allowed him to choke off all three of Rome's main grain sources, causing widespread famine in the city; Cassius Dio laconically refers to "many deaths" by hunger (48.18.1). Eventually the situation became so dire that Octavian was forced to (temporarily) recognize Sextus at the Pact of Misenum (39 BCE).
When the system was working normally, and after Egypt was added to the regular sources of supply, famine (as opposed to a less threatening grain shortage) was relatively infrequent in Rome. It was, however, always a threat, since the grain fleets could be delayed by poor weather and the amount of grain available could be attenuated by serious shortfalls in the supply regions. According to Suetonius:
"When there was a scarcity of grain because of long-continued droughts, [the emperor Claudius] was once stopped in the middle of the Forum by a mob and so pelted with abuse and at the same time with pieces of bread, that he was barely able to make his escape to the Palace by a back door; and after this experience he resorted to every possible means to bring grain to Rome, even in the winter season. To the merchants he held out the certainty of profit by assuming the expense of any loss that they might suffer from storms, and offered to those who would build merchant ships large bounties..." (18.2)
A million people need a lot of grain.
/u/toldinstone refers to shortages caused by an interruption of the overseas supply of food into the city, but that doesn't really answer the question. The logic is that since the city could not supply itself from its own hinterland, when the overseas grain supply was interrupted there was food shortage in the city. That's true, but it does nothing to explain why overseas supply was so important. More importantly, not all food shortages in the city were due to interruptions of the outside food supply, and it in dealing with food shortage in the city we should be absolutely sure to understand how the food supply to the city worked.
In the very first place, we must differentiate between "real" shortage and artificial shortage. The former includes famine caused by bad harvests and the interruption of supply. Without exception, large ancient cities were parasitic on their hinterlands for food, and past a certain point were reliant on supply beyond the immediate hinterland. For most large cities this involved looking potentially quite far afield--Athens was supplied by Black Sea grain, and Rome primarily by the Sicily-Sardinia-Africa triangle. Sea trade was simply faster, cheaper, and often (particularly after the first century or so, at which point the number of Mediterranean shipwrecks explodes) safer. Enormous amounts of grain could be shipped from Africa for a fraction of the price and at a much faster speed than smaller amounts of grain from Campania or Apulia in Italy. A handful of large cities, like Carthage or Alexandria, could be supplied from fairly close by, but this is typically due to the presence of grain-producing areas not far from their hinterlands and--most importantly--waterways that could be exploited. In the case of Alexandria, the Nile was an easy transport route, and the African grain-producing areas were coastal and only a short hop away from Carthage (Gil Gambash's contribution to the recent workshop volume The Impact of Mobility and Migration in the Roman Empire lays out the reasons for thinking that most Mediterranean trade consisted of short-range sea voyages).
Together with this must be considered the realities of agriculture in the ancient world. Even in highly fertile regions the possibility of catastrophic, failed harvests in a pre-Green Revolution agrarian world was high, sudden, and acute. Entire provinces could be devastated, and there was no way to predict when the situation would change. This is important because the supply to the city was not the sole, or even necessarily the most important, reason for "artificial" shortages of food. The ever-changing conditions of the Mediterranean agrarian economy meant that at any given moment the relationships between who was supplying whom and with what might be unpredictable. The grain supply to the city was a market supply, and therefore one that was susceptible to market conditions. Grain ships might be literally getting interrupted by storms, piracy, or wars, and we know of plenty of shortages that occurred for this reason. Nonetheless, that does not mean that the actual supply to the city was cut off. Even before Pompey's war against the pirates, when Mithridates effectively waged proxy war on the Romans and cut off contact with the Sicily-Sardinia-Africa triangle, there was still ample actual supply. Likewise, in the aftermath of Clodius' lex frumentaria we also hear of a food shortage in the city, with nobody cutting off supply. Grain could enter the city from other sources, including from Italy itself. The price of grain, however, could fluctuate quite wildly, and even when we know that there was no real supply problem--most of the time, in fact--the food shortages are often being caused by price fluctuation.
The worst, and in the Republic the most prevalent, type of price fluctuation was price manipulation. In the case of the war against the pirates, the increased risk to grain ships meant that grain sellers and grain haulers increased their prices in order to offset the possibility that they might lose a shipment. However, at the same time, grain sellers were notorious for the massive stockpiles, to which Cicero refers, that they maintained in times of shortage and plenty alike to protect themselves from sudden market shocks. Prior to the pirate war grain sellers, while well-stocked in actual grain, were restricting the amount of grain that they were releasing from their stockpiles because to provide the grain for cheap would not be economically in their interests, and because they had little assurance as to when the market conditions would change--Cicero claims that the day that command against the pirates was voted to Pompey grain prices dropped to their pre-crisis levels instantly, as grain sellers became more confident. This is indicative of the way the grain market worked. Now, the city had maintained stockpiles of public grain, to be sold at reduced price (and after 58, distributed gratis), since the middle of the second century. The problem there is multifold. One is that these stockpiles, while enormous, were severely limited compared to what private sellers could bring to market. Moreover, while much of this public grain was gotten from provincial tribute, much of it--especially during food crises--had to be purchased by the state at market prices, as when Cato was sent to Cyprus to oversee the arrangement for providing funds to the state to offset the food shortage in 57.
Arguably more importantly, the public grain, while very prominent in the modern imagination, was by no means the primary means of supplying the city at any time. Before 58 the public grain was provided at reduced price. After 58 it was distributed free, originally to every free citizen but by the time of Augustus only to around 200,000--at its low point under Caesar the public grain was only accessible to a mere 150,000. Moreover, the public grain consisted of only 5 modii per month, an amount sufficient for an individual but not for his family. The majority of a family's food therefore had to be bought at market prices, and even though under the Principate regulations were introduced to maintain supply and costs on the private market nonetheless these were never entirely effective, as the market could change quite quickly. Even availability of public grain versus private grain could affect the grain market, and the source of grain could change prices wildly.
There's much more going on here than simply ships not getting in to port. Excellent reading on the subject may be found in Garnsey's triple works Famine and Food Supply in the Graeco-Roman World, Cities, Peasants and Food in Classical Antiquity, and Food and Society in Classical Antiquity, Rickman's The Corn Supply of Ancient Rome (1980), the relevant portions Harrison's article Catiline, Clodius, and Popular Politics at Rome During the 60s and 50s BCE (2008, really not about this topic, but provides one of the most succinct overviews of the lives of the urban poor I've ever seen), Brunt's The Roman Mob (1966, old but the basis for the "modern" view of the urban poor), and Purcell's chapter in the Cambridge Ancient History, entitled the City of Rome and the Plebs Urbana in the Late Republic (though I must disagree with many of his comments about collegia).