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It depends on the individual post office as to the length of time that money is made available to people. Some post offices only have availability for 3 days only.
No, An Post & DSP have entered into an agreement whereby An Post are authorised to process DSP payments according to instructions received from DSP.
Accordingly, An Post offices have no powers of decision as they are merely acting as a paid cash dispensing agent.
When a DSP card is swiped in a Post Office the Post Office employee must (upon being satisfied of the correct identity of the DSP client) pay in cash whatever sum appears on their screen as amount being due to the DSP client presenting themselves to them. The agreed procedure for DSP clients presenting a 'book' works in the same way.
DSP authorises An Post to dispense cash to their clients from certain dates & expire on certain dates depending on the DSP Fund that clients are being paid from.
An Post employees have absolutely no powers of veto/descretion in this matter, unless in very exceptional circumstances (e.g. following an armed robbery, during industrial disputes by secure cash delivery services etc.) when they don't physically have cash available to dispense.
If you have been refused a DSP payment during the authorised DSP timeframe by a particular An Post employee you should report the matter immediately to your local DSP office as this would be in breach of their contract with DSP.
In the majority of cases, by virtue of the fact that DSP clients are unemployed/OAP/disabled/depending on their DSP payment, they pick it up on the day that it first becomes available, but An Post are bound to pay out until it becomes unavailable. An Post have no relationship with the DSP client, apart from the physical one of literally verifying, processing & dispensing.