diff --git a/zebra-network/src/peer_set/unready_service.rs b/zebra-network/src/peer_set/unready_service.rs
index 97fc8d7e0..108a9e830 100644
--- a/zebra-network/src/peer_set/unready_service.rs
+++ b/zebra-network/src/peer_set/unready_service.rs
@@ -1,5 +1,6 @@
-// Adapted from tower-balance
-
+/// Services that are busy or newly created.
+///
+/// Adapted from tower-balance.
 use std::{
     future::Future,
     marker::PhantomData,
@@ -21,11 +22,18 @@ mod tests;
 #[pin_project]
 #[derive(Debug)]
 pub(super) struct UnreadyService<K, S, Req> {
+    /// The key used to lookup `service`.
     pub(super) key: Option<K>,
+
+    /// A oneshot used to cancel the request the `service` is currently working on, if any.
     #[pin]
     pub(super) cancel: oneshot::Receiver<CancelClientWork>,
+
+    /// The `service` that is busy (or newly created).
     pub(super) service: Option<S>,
 
+    /// Dropping `service` might drop a request.
+    /// This [`PhantomData`] tells the Rust compiler to do a drop check for `Req`.
     pub(super) _req: PhantomData<Req>,
 }
 
diff --git a/zebra-network/src/protocol/external/tests/preallocate.rs b/zebra-network/src/protocol/external/tests/preallocate.rs
index 25ca76e5f..3875e8a5a 100644
--- a/zebra-network/src/protocol/external/tests/preallocate.rs
+++ b/zebra-network/src/protocol/external/tests/preallocate.rs
@@ -1,6 +1,6 @@
 //! Tests for trusted preallocation during deserialization.
 
-use std::convert::TryInto;
+use std::env;
 
 use proptest::prelude::*;
 
@@ -20,7 +20,16 @@ use crate::{
     },
 };
 
+/// The number of test cases to use for expensive proptests.
+const DEFAULT_PROPTEST_CASES: u32 = 8;
+
 proptest! {
+    // Set the PROPTEST_CASES env var to override this default.
+    #![proptest_config(proptest::test_runner::Config::with_cases(env::var("PROPTEST_CASES")
+                                                                 .ok()
+                                                                 .and_then(|v| v.parse().ok())
+                                                                 .unwrap_or(DEFAULT_PROPTEST_CASES)))]
+
     /// Confirm that each InventoryHash takes the expected size in bytes when serialized.
     #[test]
     fn inv_hash_size_is_correct(inv in any::<InventoryHash>()) {
@@ -71,6 +80,12 @@ proptest! {
 }
 
 proptest! {
+    // Set the PROPTEST_CASES env var to override this default.
+    #![proptest_config(proptest::test_runner::Config::with_cases(env::var("PROPTEST_CASES")
+                                                                 .ok()
+                                                                 .and_then(|v| v.parse().ok())
+                                                                 .unwrap_or(DEFAULT_PROPTEST_CASES)))]
+
     /// Confirm that each AddrV1 takes exactly ADDR_V1_SIZE bytes when serialized.
     /// This verifies that our calculated `TrustedPreallocate::max_allocation()` is indeed an upper bound.
     #[test]
@@ -122,6 +137,12 @@ proptest! {
 }
 
 proptest! {
+    // Set the PROPTEST_CASES env var to override this default.
+    #![proptest_config(proptest::test_runner::Config::with_cases(env::var("PROPTEST_CASES")
+                                                                 .ok()
+                                                                 .and_then(|v| v.parse().ok())
+                                                                 .unwrap_or(DEFAULT_PROPTEST_CASES)))]
+
     /// Confirm that each AddrV2 takes at least ADDR_V2_MIN_SIZE bytes when serialized.
     /// This verifies that our calculated `TrustedPreallocate::max_allocation()` is indeed an upper bound.
     #[test]
diff --git a/zebrad/tests/acceptance.rs b/zebrad/tests/acceptance.rs
index c1fd951ce..7e1827e67 100644
--- a/zebrad/tests/acceptance.rs
+++ b/zebrad/tests/acceptance.rs
@@ -1811,6 +1811,15 @@ async fn delete_old_databases() -> Result<()> {
 
     zebra_test::init();
 
+    // Skip this test because it can be very slow without a network.
+    //
+    // The delete databases task is launched last during startup, after network setup.
+    // If there is no network, network setup can take a long time to timeout,
+    // so the task takes a long time to launch, slowing down this test.
+    if zebra_test::net::zebra_skip_network_tests() {
+        return Ok(());
+    }
+
     let mut config = default_test_config()?;
     let run_dir = testdir()?;
     let cache_dir = run_dir.path().join("state");