My 5 on standard suspension felt like that when I first got it.
It came from a dealer and had been serviced by them, so I was amazed to find that the tyre pressures were all over the place. Correcting them to the standard 29 psi all round made a big difference.
Suspension is supposed to work with the vehicle remaining relatively calm and composed, gliding along nice and level with the wheels pattering up and down , following the road surface.
Large heavy cars naturally tide well because they tend to have a good srung to unsprung weight ratio.
The vehicle is heavy with lots of mass and the wheels are relatively light, so when on a bumpy surface the wheels move up and down but the body doesn’t.
This is much harder to achieve with a light sports car, so you have to be very careful about what wheels and tyres you fit. It is very possible that your wheels and tyres are heavier than standard…
Also, short springs don’t absorb bumps as well as long ones and ultra low profile tyres don’t absorb much either.
Another problem is that convertible bodies are not as stiff as saloons, as they have no roof. This means that stiffer springs are likely to make the body flex even more than it would otherwise.
All this adds up.
Hopefully, correct tyre pressures will help but who knows what is correct for your 18" low profile tyres?
29 psi is probably a good starting point but may not be ideal.
One piece of good news is that it is possible to cheaply stiffen the body by replacing the standard loose fit steel and rubber door bushes with tight fitting machined Delrin versions.
Un my experience, these made my 17" wheeled nc2 much less skittish and squirrel.
The door bushes are on the back edge of the door frame with corresponding cups for them to fit into on the door itself
999to5 Your explanation sums it up perfectly.The car will be much better balanced with lighter 17” wheels when I had my last NC1 I fitted a set of Kosei K1 TS lightweight wheels on it only 14 lbs weight per wheel , the difference in handling was really very noticeable.
I’ve just weighed my spare Niseko 17" wheel (unique to this car, apparently) fitted with an unused Dunlop Maxx RT2 all still in the wheel’s plastic bag, and it totals almost exactly 16kg.
Returning yesterday on the wet bad surface of the A41 junction under the M25, this thread came to mind as my six-year-old tyres skittered and jiggled, flashing the dreaded DSC light.
This was the final straw, decision made; soon I’ll be visiting NickD in Northampton for five new matching tyres, and probably going to 215 wide instead of the 205.
Hi Richard yes if the tyres are 6 years old they probably are becoming less pliable with age , and therefore giving less grip that’s the problem with low mileage cars , plenty of tread left , but the rubber becomes hard over time. Going back to the original posters problems I think 18” wheels are not really suitable for MX 5s, as has been previously posted to big and to heavy these cars are very sensitive to unsprung weight especially combined with lowering springs.
Just for clarity in my earlier post I stated that Kosei K1 TS wheels weigh in at 14 lbs per wheel that’s just the wheel alone l think that the stock Mazda wheels weigh in at 17 Lbs approx.
So i could swap to 17’s with appropriate tyres and that 1" will make all the difference? 
pardon the pun
Lowering an NC will change the steering geometry and thus will need wheel alignment, there are different settings for different ride heights in the Mazda aligment info.
Also changing the ride height will of course change the angle of the steering and likely to cause more bump steer.
If ride height has changed then wheel aligment needs to be done by an expert that knows what they are doing or tramlining, bump steer and handling in general may severly impact upon steering quality.
Mazda spent a lot of time and money developing their cars for a variety of roads around the World and unless for serious track work I leave my cars as designed with respect to suspension and steering and upgrade items for improved comfort, safety and convenience and enjoy the experience.
Roy
Hi Cazmaz MX5s we’re built and designed to run on16” or 17” reasonably light wheels. As stated by other members bigger heavier wheels , although they might look good are not really the best for them , also a good 4 wheel alignment can improve things by a great amount. Hope this is of some help to you. Regards Bob.
No, it won’t make all the difference unfortunately.
It might help though.
The first thing to check is tyre pressures and start with 29 all round as a base setting.
Next thing is 4 wheel alignment.
Thanks everyone ive gone through the folder and it was lowered with Apex springs but cant see any receipts for any coil covers or wheel alignment etc. If i had not been told it was lowered i would not have known by just looking at the car. Its gonna be big dollar i expect to put it back to stock. I wont be doing big miles in the car and probably no long trips so as long as the car is safe and the alignment and tyre pressures are sorted im happy to keep it as it is until i can afford to sort it out.
Sounds fair 
Hi Carol, good to hear all of the sound advice.
I am optimistic that a combination of adjusting the tyre pressures and potentially getting the wheels aligned will resolve your issues without breaking the bank.
It seems a pity to have to replace the wheels that you find such an attractive aspect of the car.
Good luck and please keep us posted on how you get on.
Cheers,
Guy
Actually, I’ll bet you could do it for zero cost, if you wanted to.
Simply advertise a straight swap on here in exchange for doing the work .
You’d want some evidence of competence though.
It is quite a common reported issue that lowering springs with the standard dampers results in a less than desirable experience. You are asking the damper to deal with body movement over a shorter travel. Add in heavier wheels and thinner tyres (which may well be over inflated) and the suspension will struggle to control the wheel movement over bumps. I suspect the main cause of issue is the spring rate. Softer sprung cars do deal with bumps better unless the damping is top notch. If you can get a set of the OEM Bilstein dampers they do control lowered suspension better. Your tyres are Toyo T1R by the way which shouldn’t really matter, but they do have a certain wobblyness to them as well.
Seen these on flea bay would they be any good? Could i still keep the 18" wheels? Mazda MX5 MK3 NC Bilstein Suspension | eBay
As the resident tyre guru, could you recommend tyre pressures, Nick??
I would say go for it.
I would stick with 29 as you still need to support the weight of the car and if you go to low you run the risk of hitting the wheels on the edge of potholes.
I don’t think the Accelera brand is a particularly good quality tyre?
May not be the cheapest, but certainly at the lower budget end of the market…
Rob