Towards the end of the TNG episode "The Best of Both Worlds (Part 2)", we have the following dialogue concerning Picard's condition:
CRUSHER: Life signs are stable. The DNA around the microcircuit fibre implants is returning to normal.
TROI: How do you feel?
PICARD: Almost human. With just a bit of a headache.
CRUSHER: We'll get you to Sickbay. We won't have any trouble getting these implants out now.
However, six years later in Star Trek: First Contact, Picard is able to hear the Borg Collective during the Battle of Sector 001, as confirmed by @Richard's answer to this question. From the licensed novelization:
...his question had already been answered by a whisper in his own head — a voice that was one, yet many; a voice that evoked the ghost of a half-remembered feminine face.
Without thinking, he rose, entranced, and moved toward the viewscreen where the image of the massive and unlovely vessel hung. They were there; he could sense them, hear them speak. For an instant, he felt as though he had only to reach toward the screen, and he would touch them.
The whisper of the one and the multitude grew briefly louder. "...critical damage to shields at power sector 1-1-1. All drones coordinate repair immediately...."
The mental whisper died abruptly, as if the speakers had realized he was listening. But it was too late; he had already experienced a revelation beyond the mere words he had detected.
They were wounded. They were vulnerable, and he knew beyond all reason the precise spot.
How is it that Picard can still hear the Borg hive mind and / or sense the Collective's presence?
Assuming that Crusher did as she said and removed all of Picard's Borg implants and organelles, there seems to be no technological or biological way for him to intercept signals from the Collective.
To contrast with Seven of Nine, she still had several Borg implants — including a Borg subspace transceiver — and so it made sense that she could still detect the Collective's presence.