Service of the transmission is REQUIRED. From what I can find and know this appears to be 2nd Band Failure. Parts to repair this if that is the only thing wrong are not expensive.The Labor to repair it will be the largest part of the bill. [http://www.drivetrain.com/autotransproblemsolve.html Look here for details] Once there search for "All C5 / C4" Hope This Helps.
Transmission temp sensor can cause a slip or high rpm before shifting to next gear.
Quite possibly a worn second gear clutch pack.
the synchronize in the transmission is bad, that why the jumping out of fifth gear
1 A bent shifting fork from shifting hard 2 a worn bearing inside the transmission high miles or ran low of lube 3 a worn gear 4 hopefully an adjustment on the shifter
Means that the transmission is delaying going into the next gear.
Worn out gears in a transmission will cause the transmission to slip out of gear. The gear may be war only on one side, causing the transmission to work properly. At times.
If it's only when you attempt third gear the problem is in your transmission.
The synchros are going out on the tranny
Low fluid level or the clutch packs are worn out.
we would need more info on this question there a few reasons this could happen depending on the circumstances ie bad linkage detents or loose linkage internal transmission problems ect.i could be more exact if i had more info
A 1994 Mazda Miata's automatic transmission may slip in 4th gear if the bands are heavily worn. When the bands become worn, they are unable to grip properly and cause slippage.
bad linkage in shifting cable