yes and no.

Anarchist power rise because they stopped the coup on Barcelona and got an arsenal, have the hegemony among the working class on Catalonia, and among the chaos of the july they become an important armed force. They then start to blend with the Republican State, while having autobomous groups, like every party and syndicate (armed "police" patrols, militias, etc.) May of 37 marks the political decline of CNT and the rise of PSUC (soviet communists), but they have so much power between the working class (look into affiliation numbers) that they weren't able to just delete them, like they did with POUM (autonomous communists, they weren't troskysts as psuc said)

At first, some insight in economy: anarchist collectivisations were done mainly on Catalonia and the rural republican part of Aragon (and other parts of the Republic also). Some, were dissolved by communists, but a lot of them were working until the last days of the republic. All of thid was made legal by catalan republix authorities, altough the tendency was to gradually retake the power and control of the situation, but there was still some council decision made by workers (and possibily a rising anarchist burocracy)

So, with all of this you have police, army, politicians and even jails runned by anarchist. You can say that they were traitors by joining the republicans, and they were attacked by communist traitors (main views of Federica Montseny and the mainly view of anarchism that survived into the exile and later come back as main CNT views).

Or you can see that anarchism, faced the war as it could, allied the republicans but pushed a big revolution, but not the big anarchocommunist one). They merged the State, and they autonomous police and militia were a proto-State. It could have evolved into a Syndicalist Republic. This view is not popular, but I think that fits more the reality as a sociological POV.

My personal view is that anarchism, in the practive, is not that different to communism. It potentially have more worker-making decisions on the economy, but Titoism have some degree of that, based on the CNT experience that he saw when he was in Spain fightin in the International Brigades.

That's my personal view of everything. There's a lot of books regarding the experiencies of the CNT people that joined the government and made his personal review, and also there's about collectivisations.

Sources that maybe you should read

Politicians of CNT:Anarchists in the Spanish Revolution, Vuitanta dies al govern de la Generalitat (that would be cool to have in english)

Collectivisations: With the Peasants of Aragon, Frank Mintz works, Collective Economy. Europe's last revolution (documentary, not a book)