Your suggestion is not impossible and difficult to rule out, but I can offer a couple of data points.
During the events of Harry Potter and the Order of the Phoenix harry receives lots and lots of mail in response to his Quibbler interview while Dumbledore was still headmaster.
She gave the delivery owl a Knut and unfolded the newspaper eagerly while Harry helped himself to orange juice; as he had only received one note during the entire year, he was sure, when the first owl landed with a thud in front of him, that it had made a mistake.
'Who're you after?' he asked it, languidly removing his orange juice from underneath its beak and leaning forwards to see the recipient's name and address:
Frowning, he made to take the letter from the owl, but before he could do so, three, four, five more owls had fluttered down beside it and were jockeying for position, treading in the butter and knocking over the salt as each one attempted to give him their letter first.
'What's going on?' Ron asked in amazement, as the whole of Gryffindor table leaned forwards to watch and another seven owls landed amongst the first ones, screeching, hooting and flapping their wings.
Harry Potter and the Order of the Phoenix - pp.509-10 - Bloomsbury - Chapter 26, Seen and Unforeseen
Not all of this mail is nice.
'This one's from a bloke who thinks you're off your rocker,' said Ron, glancing down his letter. 'Ah well ...'
'This woman recommends you try a good course of Shock Spells at St Mungo's,' said Hermione, looking disappointed and crumpling up a second.
Harry Potter and the Order of the Phoenix - p.511 - Bloomsbury - Chapter 26, Seen and Unforeseen
It is also noteworthy that Dumbledore does not stop Harry from receiving his Firebolt, even though he still believed Sirius a criminal at that point and it arrived in very suspicious circumstances.
'May I?' said Professor McGonagall, but she didn't wait for an answer before pulling the Firebolt out of their hands. She examined it carefully from handle to twig-ends. 'Hmm. And there was no note at all, Potter? No card? No message of any kind?'
'No,' said Harry blankly.
'I see ...' said Professor McGonagall. 'Well, I'm afraid I will have to take this, Potter.'
'W-what?' said Harry, scrambling to his feet. 'Why?'
'It will need to be checked for jinxes,' said Professor McGonagall.
'What did you go running to McGonagall for?'
Hermione threw her book aside. She was still pink in the face, but stood up and faced Ron defiantly.
'Because I thought - and Professor McGonagall agrees with me - that that broom was probably sent to Harry by Sirius Black!'
Harry Potter and the Prisoner of Azkaban - pp.171-2 - Bloomsbury - Chapter 11, The Firebolt