Getting a permanent good fit on a box with a push fit lid can be tricky. As Cornucopia says, it is important to rough turn the box and let it "rest" for a long as possible before finish turning - and it should ideally be rested in the environment it will eventually be kept when it is finished (ie probably a centrally heated house).
This is what I do when I make boxes with push fit lids....
1. Rough turn the blank - which means make it into a cylinder, cut chucking spigots on either end, part off lid from base, then rough hollow the two parts - do not leave the walls too thick at this stage, but leave enough material for trueing up should the blank go hollow (this is less likely with a dry blank, very likely with a wet log!).
2. Tape the two parts together and bring indoors. Weigh the parts and note the weight on them. If the wood was already "dry" I then put the rough turned blank in the airing cupboard (to simulate warm, dry centrally heated house - which mine isn't) and leave it there until the weight stops dropping. The wood is then very, very dry.
3. Take the blanks out to the workshop and finish the box - do not leaave the blank in the workshop unless you are working on it (ie bring it indoors between stages if you don't have time to finish it all in one go.)
4. Actually fitting the lid - (for an over fitting lid). I finish and sand the inside of the lid - making sure the part where the flange of the base will fit is absolutely parallel. Then I mount the base on the lathe and finish & sand the inside of the box - the reason for this is because finishing the inside *may* cause further movement of the wood and I want that out the way before fitting the lid.
Next, I cut the flange for the lid. Initially this is cut as a slight taper (so the lid goes on part way and leaves a witness mark.) Then the flange is then cut parallel to the size left by the witness mark. I tend to use a small parting tool for this, but angled so it's actually cutting like a skew chisel and leaves a nice clean surface that would need no sanding. The lid should now fit VERY tightly - in engineering terms "a tight fit". I now finish & sand the outside of the box and the top of the lid (with tailstock support until I absoluely have to remove it.) Which leaves a finished, smoothly sanded box with the lid stuck on really tight - I find wearing a leather glove helps grip the lid so I can now remove it to finally fit the lid. Again, using a small parting tool as a skew, just kiss the surface of the wood and keep stopping and checking the lid fit as you go until you are happy with it. At this stage you could also shape the flange so it is very slightly barrel shaped to get a nice suction fit - Richard Raffan's book on making boxes goes into flange shapes in great detail, BTW and I found it really useful when I first started making them.
5. Having got the box to this stage - I take it off the lathe and back in doors - with the chucking spigot still on the base and leave for another couple of days. If the lid goes too tight - it can be remounted and dressed very carefully with fine sandpaper (400 grit or finer) until it fits again. If it goes the other way you can sometimes recover a slightly loose lid by applying an extra coat or two of whatever finish you use around the flange area. I use hardwax oil on all my boxes which is applied off the lathe so this is not a problem.
6. Some finishes and colouring techniques will affect the fit of the lid - any finish that raises the grain of the wood will make the lid just a bit tighter. Wax on the flange may make a tight lid a bit easier to get on or off. Adding texture bands to lid and base makes a box easier to grip for removal of tight lids.
7. Some woods move more than others which can be tricky for boxes. I've found that walnut hardly moves at all, whereas boxwood moves a lot more than you would expect!
8. Large boxes (3" or more in diameter) cause more problems with movement than small boxes - large boxes with tight fitting lids can be a nuisance to open - I usually make large boxes with looser fitting lids, or with threaded lids.
9. Make sure any finishes you use have cured properly before leaving the box with it's lid on. Leave at least 24 hours to be sure. Acrylic finishes may "stick" on mating parts if not properly cured, even though the finish feels "dry" to the touch.