diff --git a/.github/workflows/coverage.yml b/.github/workflows/coverage.yml
index 3ab62d80eb44..7bef242ba019 100644
--- a/.github/workflows/coverage.yml
+++ b/.github/workflows/coverage.yml
@@ -5,7 +5,9 @@ on:
     branches: ['main']
 concurrency:
   group: ${{ github.repository }}-${{ github.ref }}-${{ github.head_ref }}-${{ github.workflow }}
-  cancel-in-progress: true
+  # Only cancel in PR mode.  In push mode, don't cancel so we don't see spurious test "failures",
+  # and we get coverage reports on Coveralls for every push.
+  cancel-in-progress: ${{ github.event_name == 'pull_request' }}
 jobs:
   coverage:
     name: Coverage
diff --git a/.github/workflows/neko.yml b/.github/workflows/neko.yml
index 74b697d2533a..9af0ced078ba 100644
--- a/.github/workflows/neko.yml
+++ b/.github/workflows/neko.yml
@@ -5,7 +5,8 @@ on:
     branches: ['main', 'stable/*']
 concurrency:
   group: ${{ github.repository }}-${{ github.ref }}-${{ github.head_ref }}-${{ github.workflow }}
-  cancel-in-progress: true
+  # Only cancel in PR mode.  In push mode, don't cancel so we don't see spurious test "failures".
+  cancel-in-progress: ${{ github.event_name == 'pull_request' }}
 jobs:
   neko:
     name: Qiskit Neko Integration Tests