diff --git a/zebra-network/src/peer_set/initialize.rs b/zebra-network/src/peer_set/initialize.rs
index 6822fc25..1e0feb66 100644
--- a/zebra-network/src/peer_set/initialize.rs
+++ b/zebra-network/src/peer_set/initialize.rs
@@ -260,14 +260,34 @@ where
             }
             Err((addr, ref e)) => {
                 handshake_error_total += 1;
+
                 // this is verbose, but it's better than just hanging with no output when there are errors
-                info!(
-                    ?handshake_success_total,
-                    ?handshake_error_total,
-                    ?addr,
-                    ?e,
-                    "an initial peer connection failed"
-                );
+                let mut expected_error = false;
+                if let Some(io_error) = e.downcast_ref::<tokio::io::Error>() {
+                    // Some systems only have IPv4, or only have IPv6,
+                    // so these errors are not particularly interesting.
+                    if io_error.kind() == tokio::io::ErrorKind::AddrNotAvailable {
+                        expected_error = true;
+                    }
+                }
+
+                if expected_error {
+                    debug!(
+                        successes = ?handshake_success_total,
+                        errors = ?handshake_error_total,
+                        ?addr,
+                        ?e,
+                        "an initial peer connection failed"
+                    );
+                } else {
+                    info!(
+                        successes = ?handshake_success_total,
+                        errors = ?handshake_error_total,
+                        ?addr,
+                        %e,
+                        "an initial peer connection failed"
+                    );
+                }
             }
         }
 
diff --git a/zebra-network/src/peer_set/limit.rs b/zebra-network/src/peer_set/limit.rs
index 0d2ff1af..6c2f1849 100644
--- a/zebra-network/src/peer_set/limit.rs
+++ b/zebra-network/src/peer_set/limit.rs
@@ -105,7 +105,7 @@ impl ConnectionTracker {
     fn new(counter: &mut ActiveConnectionCounter) -> Self {
         counter.count += 1;
 
-        info!(open_connections = ?counter.count, "opening a new peer connection");
+        debug!(open_connections = ?counter.count, "opening a new peer connection");
 
         Self {
             close_notification_tx: counter.close_notification_tx.clone(),
diff --git a/zebra-state/src/service.rs b/zebra-state/src/service.rs
index ab3713c8..1ee750f4 100644
--- a/zebra-state/src/service.rs
+++ b/zebra-state/src/service.rs
@@ -445,7 +445,7 @@ impl StateService {
         let chain_tip_height = if let Some((height, _)) = self.best_tip() {
             height
         } else {
-            tracing::info!(
+            tracing::debug!(
                 response_len = ?0,
                 "responding to peer GetBlocks or GetHeaders with empty state",
             );
diff --git a/zebrad/src/components/inbound.rs b/zebrad/src/components/inbound.rs
index d5f2d393..f06b3f39 100644
--- a/zebrad/src/components/inbound.rs
+++ b/zebrad/src/components/inbound.rs
@@ -372,7 +372,8 @@ impl Service<zn::Request> for Inbound {
                         .map_ok(|_resp| zn::Response::Nil)
                         .boxed()
                 } else {
-                    info!(
+                    // Peers send a lot of these when we first connect to them.
+                    debug!(
                         "ignoring `AdvertiseTransactionIds` request from remote peer during network setup"
                     );
                     async { Ok(zn::Response::Nil) }.boxed()
@@ -385,7 +386,8 @@ impl Service<zn::Request> for Inbound {
                 {
                     block_downloads.download_and_verify(hash);
                 } else {
-                    info!(
+                    // Peers send a lot of these when we first connect to them.
+                    debug!(
                         ?hash,
                         "ignoring `AdvertiseBlock` request from remote peer during network setup"
                     );