First, it is US Citizenship and Immigration Services (not Customs).
Second, are C-files likely to have anything interesting in them? The problem is, as always, it depends on the date and the individual case. You may hear from one person they got a C-file with a wealth of information, and from another that the file they received had only duplicate copies of the same documents found in court records (except the certificate, more on that below). Both are telling you the truth.
C-Files for naturalizations after WW II and to 1956 are usually complete, containing all the records maintained by INS about that subject. Earlier C-files, from 1906 to the end of WW II, should only contain the naturalization records and any additional documents related to that naturalization. If the process went smoothly for the immigrant in this earlier period the file likely contains only the duplicate declaration, duplicate petition, and certificate of naturalization. But if there was any issue (about fees, qualifications, problems getting a certificate of arrival, etc., etc.,) there could be additional documentation, including forms and correspondence. Also, if the naturalized citizen later applied for a replacement certificate, or if a wife or child derived US citizenship through their naturalization and later applied for a derivative certificate, there would be additional records. Those additional records, if such exist, can likely be found no where else.
Unless you have reason to believe there are additional records in an immigrant's C-file, there is only one good way to predict if the C-file has more records. You can search the Name Index to Bureau of Naturalization Correspondence, 1906-1946 (National Archives microfilm publication A3388). If correspondence was placed in the C-file, it should be indexed there and reference the C-file (certificate) number. Not finding a name in this index does not prove there are no extra records in the C-file, but finding the name in the index does indicate there is something more.
Third, the only document in every C-file that is not found in court records is the duplicate certificate of naturalization. There are many post-1906 court naturalization records online somehow classified or tagged as "certificates." I often see naturalization index cards identified on Ancestry as "certificates" but of course they are not. They are index cards.