Is there an advantage to storing keys split between several hashes?
Lets take your idea to the extreme to see its weakness. For simplicity I'll scale things down. Let's assume an 8 bit key chosen randomly. Call this key $k$. If I break into your database and get $d=H(k,s)$ along with $s$ (where $H$ is a hash function and $s$ is the salt), it would take on average $2^{8-1}=2^7=128$ computations (or calls to $H$) to find $k$.

Now, using your idea, lets say the server instead splits $k$ into $(b_1,b_2,\cdots,b_8)$ (i.e., bits) and then computes $d_i=H(k_i,s_i)$ (so there is a different salt for each bit of the key). Now, the attacker who breaks into your database and steals the pairs $(d_i, s_i)$ needs to crack all $8$ hashes. More difficult? No.

The reason is, how much work does it take to break $d_1$? Well, the attacker only need compute $H(1,s_1)$ and $H(0,s_1)$ to find $b_1$. That is only two operations (one on average though since half of the time the attacker will get the right value on the first try). Do this for all $8$ bits and the attacker has to make a maximum of $16$ hash function calls to break the key $k$. The average number of hash function calls would be $12$. This is much, much less than in the normal case above.

Asymptotically speaking, the first system (which hashes the entire password) is something like $O(2^n)$ where $n$ is the number of bits in the key and the second system (your proposal) is $O(n)$.