diff --git a/jsonnet/mimir-mixin/alerts/compactor.libsonnet b/jsonnet/mimir-mixin/alerts/compactor.libsonnet
index be3de8c0c2..1f28a7e54a 100644
--- a/jsonnet/mimir-mixin/alerts/compactor.libsonnet
+++ b/jsonnet/mimir-mixin/alerts/compactor.libsonnet
@@ -47,6 +47,19 @@
             message: 'Cortex Compactor {{ $labels.namespace }}/{{ $labels.instance }} has not run compaction in the last 24 hours.',
           },
         },
+        {
+          // Alert if compactor failed to run 2 consecutive compactions.
+          alert: 'CortexCompactorHasNotSuccessfullyRunCompaction',
+          expr: |||
+            increase(cortex_compactor_runs_failed_total[2h]) >= 2
+          |||,
+          labels: {
+            severity: 'critical',
+          },
+          annotations: {
+            message: 'Cortex Compactor {{ $labels.namespace }}/{{ $labels.instance }} failed to run 2 consecutive compactions.',
+          },
+        },
         {
           // Alert if the compactor has not uploaded anything in the last 24h.
           alert: 'CortexCompactorHasNotUploadedBlocks',
@@ -65,7 +78,7 @@
         },
         {
           // Alert if the compactor has not uploaded anything since its start.
-          alert: 'CortexCompactorHasNotUploadedBlocksSinceStart',
+          alert: 'CortexCompactorHasNotUploadedBlocks',
           'for': '24h',
           expr: |||
             thanos_objstore_bucket_last_successful_upload_time{job=~".+/%(compactor)s"} == 0
@@ -77,21 +90,6 @@
             message: 'Cortex Compactor {{ $labels.namespace }}/{{ $labels.instance }} has not uploaded any block in the last 24 hours.',
           },
         },
-        {
-          // Alert if compactor fails.
-          alert: 'CortexCompactorRunFailed',
-          expr: |||
-            increase(cortex_compactor_runs_failed_total[2h]) >= 2
-          |||,
-          labels: {
-            severity: 'critical',
-          },
-          annotations: {
-            message: |||
-              {{ $labels.job }}/{{ $labels.instance }} failed to run compaction.
-            |||,
-          },
-        },
       ],
     },
   ],
diff --git a/jsonnet/mimir-mixin/docs/playbooks.md b/jsonnet/mimir-mixin/docs/playbooks.md
index cc3a3ad928..704b649282 100644
--- a/jsonnet/mimir-mixin/docs/playbooks.md
+++ b/jsonnet/mimir-mixin/docs/playbooks.md
@@ -272,11 +272,21 @@ Same as [`CortexCompactorHasNotSuccessfullyCleanedUpBlocks`](#CortexCompactorHas
 This alert fires when a Cortex compactor is not uploading any compacted blocks to the storage since a long time.
 
 How to **investigate**:
-- If the alert `CortexCompactorHasNotSuccessfullyRun` or `CortexCompactorHasNotSuccessfullyRunSinceStart` have fired as well, then investigate that issue first
+- If the alert `CortexCompactorHasNotSuccessfullyRunCompaction` has fired as well, then investigate that issue first
 - If the alert `CortexIngesterHasNotShippedBlocks` or `CortexIngesterHasNotShippedBlocksSinceStart` have fired as well, then investigate that issue first
 - Ensure ingesters are successfully shipping blocks to the storage
 - Look for any error in the compactor logs
 
+### CortexCompactorHasNotSuccessfullyRunCompaction
+
+This alert fires if the compactor is not able to successfully compact all discovered compactable blocks (across all tenants).
+
+When this alert fires, the compactor may still have successfully compacted some blocks but, for some reason, other blocks compaction is consistently failing. A common case is when the compactor is trying to compact a corrupted block for a single tenant: in this case the compaction of blocks for other tenants is still working, but compaction for the affected tenant is blocked by the corrupted block.
+
+How to **investigate**:
+- Look for any error in the compactor logs
+  - Corruption: [`not healthy index found`](#compactor-is-failing-because-of-not-healthy-index-found)
+
 #### Compactor is failing because of `not healthy index found`
 
 The compactor may fail to compact blocks due a corrupted block index found in one of the source blocks:
@@ -301,18 +311,6 @@ To rename a block stored on GCS you can use the `gsutil` CLI:
 gsutil mv gs://BUCKET/TENANT/BLOCK gs://BUCKET/TENANT/corrupted-BLOCK
 ```
 
-### CortexCompactorHasNotUploadedBlocksSinceStart
-
-Same as [`CortexCompactorHasNotUploadedBlocks`](#CortexCompactorHasNotUploadedBlocks).
-
-### CortexCompactorHasNotSuccessfullyRunCompaction
-
-_TODO: this playbook has not been written yet._
-
-### CortexCompactorRunFailed
-
-_TODO: this playbook has not been written yet._
-
 ### CortexBucketIndexNotUpdated
 
 This alert fires when the bucket index, for a given tenant, is not updated since a long time. The bucket index is expected to be periodically updated by the compactor and is used by queriers and store-gateways to get an almost-updated view over the bucket store.