If you're looking for a meeting place or a good stop around ½ or halfway between North Platte and Boulder, you can try Sterling, CO, which is almost 2 hours west of North Platte, or about 2 hours east of Boulder.
Another possible place to stop near the midpoint is Atwood.
About ¼ or a fourth of the way, you can stop at Beals Trailer Court, NE. This is almost an hour west of North Platte.
About ⅓ or a third of the way, you can stop at Ovid, CO. This is a little over 1 hour west of North Platte.
About ⅔ or two thirds of the way, you can stop at Fort Morgan, CO. This is about 2½ hours west of North Platte, or about 1½ hours east of Boulder.
About ¾ or three fourths of the way, you can stop at Masters, CO. This is about 3 hours west of North Platte, or about 1 hour east of Boulder.
If you drive straight through without stopping at all, the total distance is 277 miles. The distances shown above don't include local roads to get into each town, so if you're planning a long road trip, make sure you factor in traffic and road conditions, rest stop breaks, and interesting detours along the way. If you're in a rush, you can do the trip with zero stops, and your total travel time would be around 3 hours, 55 minutes.