If all your options are in the same month, and there's lots of time remaining, it's easy to stay gamma and delta neutral. With lots of time remaining, you can be long one strike and short a nearby strike, and as the underlying moves from one strike to the other, your gammas don't change much. So if you start out delta and gamma neutral, you remain delta and gamma neutral.
But as you approach expiration, the gamma relationship between strikes begins to break down.
An example: If you're long the 100 strike and short the 105 strike 1 to 1, and there's a year remaining, it doesn't much matter if the underlying is at 100 or 105. Either way you'll remain fairly gamma neutral. So your delta position won't change much.
But if there's little time remaining until expiration and the underlying goes to 105, you're going to become very short gammas, which will throw off your deltas. If the underlying moves to 100, you'll become very long gammas, also throwing off your deltas. You'll find it very difficult to remain gamma neutral, and therefore very difficult to remain delta neutral. You'll find yourself having to adjust often, with unpredictable results.
Also, if all your options have the same expiration, then gamma neutral pretty much equals theta neutral and vega neutral. But if you have options with different expirations (calendar spreads), that relationship breaks down completely. That's because vega and theta have an opposite relationship with time. With a lot of time remaining, vegas are high and thetas low. With little time remaining theta is high and vegas are low.