A family of Microsoft relational database management systems designed for ease of use.
Try doing a compact and repair on the database containing the tables. In the past that has taken care of the problem for me.
Access has 255 slots for fields in a table, but it uses up a slot if you redefine a field or add a field and then delete the field. It does not recover the no-longer-used slots until you do a compact and repair. So you can see 32 fields in the table, but you may have used up all the slots by changing field definitions (especially field types) and adding and then dropping fields.
I have to say that I have not run into this problem in years, but then I don't often redefine fields and my tables are almost all less then 20 fields.