There's a lot of variables not specified here - what region(s), total mileage, how often is "every now and then" for hotels, restaurants, cities, and towns. I think you absolutely could do a 4 month trip on that budget, but it would involve a lot of planning and limit your options - some combination of a lot of free/primitive camping, low total mileage, relying mostly on annual park pass and free activities for entertainment, sticking to cheap areas of the country, etc. (For some perspective, in many areas just getting a campsite could be half or all of your daily budget). So I have a sense your budget could get you a nomadic experience of some kind, but not everything you're looking for.
I'm gonna say probably not?
There's a lot of variables not specified here - what region(s), total mileage, how often is "every now and then" for hotels, restaurants, cities, and towns. I think you absolutely could do a 4 month trip on that budget, but it would involve a lot of planning and limit your options - some combination of a lot of free/primitive camping, low total mileage, relying mostly on annual park pass and free activities for entertainment, sticking to cheap areas of the country, etc. (For some perspective, in many areas just getting a campsite could be half or all of your daily budget). So I have a sense your budget could get you a nomadic experience of some kind, but not everything you're looking for.