It's possible, just not recommended. At best you'll get to JFK 1 hour before your departure, which should be okay if you aren't checking bags. But if there is traffic (which there usually is at that time of day), you're screwed!

a) Planes landing at LGA at 4PM are likely to be delayed.

b) Cabs change shifts at 4PM, which makes getting a cab at that time extremely difficult, and overloads the car service companies.

c) At rush hour, you'd be very lucky to get from one airport to the other in an hour.

d) JFK security lines are generally crowded at 5PM.

So, if you have no checked luggage, and land on time, and are sitting near the front of the plane, and miraculously get a cab quickly, and there's no traffic or construction or accidents, you MIGHT get to JFK by the skin of your teeth in time to make your gate. But I would bet a thousand dollars that you'd miss your flight.

If you have checked baggage, it isn't physically possible, no matter how lucky you get.

If you had a plane scheduled to land at 4 PM at JFK and catch a 6 PM flight from JFK (same airport) on another airline, getting your luggage then going from one terminal to another, rechecking your luggage 25 minutes before the plane leaves (1 hour if out of the country) and going through security at a second terminal, you'd have trouble doing that!

So adding in a cab ride makes it pretty much impossible.