Is this "layered" XOR cipher secure?
If I understand right, your operation effectively is
$$\forall i:  c_i = p_i \oplus k_0 \oplus k_1 \oplus k_2 \oplus \dots \oplus k_n,$$
whith $c_i$ the ciphertext bits, $p_i$ the plaintext bits, and $k_j$ the key bits.

As $\oplus$ (this is XOR) is associative, this is equivalent to 

$$k^* :=  k_0 \oplus k_1 \oplus k_2 \oplus \dots \oplus k_n,$$
$$\forall i:  c_i = p_i \oplus k^*.$$

Now $k^*$ is either 0 or 1 - in the first case this operation is the identity, in the second case it is simple negation. Neither of those can be considered an encryption, and you effectively have a key space of 2 (= 1 bit).

Whoever told you this either wanted to play a joke with you, or knows nothing of cryptography (or both). Or I misunderstood your presention of the algorithm.