Can a court give me more penalty points than the fixed penalty offered?

That is one reason to distinguish a fixed penalty from a guaranteed maximum outcome. For example, nidirect lists speeding at three to six points in the penalty guidance, even though a standard fixed-penalty speeding case may involve three points.

A court can also impose fines, costs and, for some offences, disqualification according to the legislation and circumstances.

This does not mean a driver should never contest a ticket. It means the decision to request court should be based on the evidence and a genuine dispute, with awareness of the possible sentencing range if convicted.

Penalty points, fixed penalties, court fines and disqualification are separate legal concepts. The effect depends on the offence, how the case is disposed of and the driver's existing record, so the document and offence code matter.