No. Usually the Kerckhoff principle is considered. That means that the security of a cipher should rely on the key alone. The algorithm itself should be considered known. You could think of the algorithm as a single key that anybody that knows the algorithm possesses. Such a key should not be considered secure.
There is no generic way of testing the security of an unknown cipher, even if the plaintext and ciphertext are known. A block cipher is supposed to be a PRP - a pseudo random permutation - where the specific permutation is chosen using the key. Of course you could find relationships between the plaintext and the ciphertext for very weak ciphers. Usually attacks are on specific design or implementation details of the cipher, not directly on the ciphertext.
Basically to test the cipher you should create a paper formally describing it. The paper should indicate why you think the cipher is secure. You'd then have to find experts to take a look at it and possibly attack it. The best way to do that is to enter it into a competition.
If you're just experimenting, try to study attacks on other ciphers first. Then try and perform similar attacks on both your and other ciphers. See why other ciphers are broken.
If you subject your cipher to a competition prepare to be tossed out in the first round. Remember: the old ciphers of pioneers such as Bruce Schneier and Ron Rivest would be tossed out in the first round as well.