Can I connect to any neutral wire in the house?
According to 2011 National Electrical Code article 300.3(B), No.  For verbiage, see this answer.  For one explanation why, see this answer.
Another reason not to do it, is because you could end up overloading the grounded (neutral) conductor. Depending on if the two circuits are on the same branch circuit, different branch circuits on the same leg, or different branch circuits on different legs, you'll run into different situations.  However, in two out of three of these situations, it is very easy to overload the grounded (neutral) conductor.
Examples:
Let's examine what happens in a 120/240V single split-phase system, in a few different configurations. The circuit breakers in these examples is a combination circuit breaker, with 15 ampere thermal overcurrent protection and magnetic short-circuit protection.
Load on a circuit

With a single branch circuit supplying a 15 ampere load, we see that there will be 15 amperes on the grounded (neutral) conductor.  This is fine, since the grounded (neutral), and ungrounded (hot) conductors are both sized to carry 15 amperes.
Two loads on the same branch circuit

In this situation, we'll see 30 amperes on the grounded (neutral). However, since we're also drawing 30 amperes through a single breaker with 15 ampere overcurrent protection, the breaker will open.
Two loads on the same leg, but different breakers

This is the dangerous situation, since the breakers will not trip. Each circuit will draw 15 amperes on the ungrounded (hot) conductor, but the grounded (neutral) conductor will see 30 amperes.  Since the conductor is only rated for 15 amperes, the conductor is in danger of overheating and starting a fire.
Two loads on different legs

If the two loads are on different legs. You'll find that while each ungrounded (hot) conductor sees 15 amperes, the grounded (neutral) carries 0 amperes.