The short answer is yes, and they weren't exactly wrong either.
Here is an example of a Chicago Tribune article from 1960, talking about Communist infiltration of both the Guantanamo bay base and (in the column directly to the right) of how Cuban agents had discovered and revealed plans for an invasion of Cuba by troops training in Guatemala (which is vehemently denied in the article but was 100% true, turning into the infamous 'Bay of Pigs Invasion').
Jim Rasenberger writes in The Brilliant Disaster (book about the Bay of Pigs invasion, 1961) about how Cuban agents in the midst of the exile community successfully discovered and communicated plans for invasion to Havana. Although, it should be noted, that many of the exiles participating in the invasion lacked... Discretion. It really didn't take master spies to figure out that something was in the works, especially with so many key exile figures 'disappearing' from Miami and their family members gossiping about how the liberation of Cuba would come soon.
Later, during the Mariel boatlift (1980) there were one former Cuban agent, Mario Estevez Gonzalez, claimeven testified to congress that as many as seven thousand Cuban agents had infiltrated the US during the Mariel boatlift (1980), however there doesn't seem to be much evidence for such an exorbitant claim. These kinds of claims, however, are fairly common for the period.
http://www.nytimes.com/1983/05/01/us/cuban-ties-boatlift-to-drug-trade.html
The ironic part is that Cuba has definitely infiltrated the US on numerous occasions.
Ana Montes had significant security clearance as an analyst at the DIA. She was a spy, admitted passing sensitive and classified information to Cuba, and is currently serving a rather extended sentence. While not strictly 'spies' in the traditional sense (given that there is no evidence that they infiltrated or attempted to infiltrate the US government or acquire sensitive information from any US agency), the so called Cuban Five were arrested in 1998 after having infiltrated Cuban American groups opposed to the Cuban Revolution, and in particular those with connections to paramilitary organizations which had executed numerous terrorist attacks against Cuba in the preceding years. They were only a part of a greater spy network known as the Red Avispa (Wasp Network).
The US has, of course, also sent agents to infiltrate Cuba. The US government seems to have employed the personal assistant to Parliamentary President Ricardo Alarcón (Miguel Álvarez) as a spy for decades. Álvarez was sentenced to 30 years and is perhaps the most damaging American spy in the history of the Cuban Revolution (I met both and Álvarez was actually Alarcón's drinking buddy and confidant in addition to personal assistant).
http://elcomercio.pe/mundo/actualidad/espia-liberado-cuba-agente-doble-cia-noticia-1780542
Cuba's intelligence services are famously capable and every few years they will reveal a couple of their best agents (often in high positions within the opposition or ex-pat anti-Castro groups) who then give testimony against their former anti-Castro colleagues.