I assume you're talking about a space-time manifold. If you're not, this reply may not be useful.
Christoffel symbols do vanish in a Riemann Normal coordnate system at some event labelled p. In space-time, p is an event, so it is one particular location at one particular instant of time. At events in the neighborhood of p, the Christoffel symbols are "small", but not necessarily zero. Thus, if you consider an arbitrary point q in the space-time manifold, q must be close to p both in space and in time for the Christoffel symbols to vanish.
Perhaps more usefully, in Fermi Normal coordinates, Christoffel symbols can be made to vanish near some geodesic curve. Which is what I think you might have been asking about when you asked
Riemann normal coordinates won't make the Christoffel symbols vanish along a curve, but Fermi-Normal coordinates can make them vanish along the whole worldline, as long as the worldine is a geodesic. If the worldline is not a geodesic, not all of the Christoffel symbols can be made to vanish by this coordinate choice. In some world-tube "near" the geodesic worldline, the Christoffel symbols will be small.
Going back from a discussion of space-time to a discussion of space, it's useful to note that the Christoffel symbols are coordinate dependent. If you have a flat plane with cartesian coordinates (x,y), the line element for the metric is dx^2 + dy^2 and the Christoffel symbols vanish everywhere. But if you use plolar coordinates (r,theta) for the plane, the Christoffel symbols do not vanish. The same geometry, a plane, has different Christoffel symbols depending on your coordinate choice. So asking about the symbols vanishing doesn't make sense unless you include information about the coordinates you are using.