diff --git a/go.mod b/go.mod
index 2c5b1365..d323b5c0 100644
--- a/go.mod
+++ b/go.mod
@@ -5,7 +5,7 @@ require (
 	contrib.go.opencensus.io/exporter/prometheus v0.1.0
 	contrib.go.opencensus.io/exporter/stackdriver v0.13.4
 	github.com/Azure/azure-sdk-for-go v36.2.0+incompatible // indirect
-        github.com/Azure/go-autorest v13.3.0+incompatible // indirect
+	github.com/Azure/go-autorest v13.3.0+incompatible // indirect
 	github.com/Azure/go-autorest/autorest/adal v0.8.0 // indirect
 	github.com/Azure/go-autorest/autorest/to v0.3.0 // indirect
 	github.com/Azure/go-autorest/autorest/validation v0.2.0 // indirect
@@ -48,6 +48,7 @@ require (
 	github.com/samuel/go-zookeeper v0.0.0-20190801204459-3c104360edc8 // indirect
 	github.com/spaolacci/murmur3 v1.1.0 // indirect
 	github.com/spf13/pflag v1.0.5 // indirect
+	github.com/stretchr/testify v1.4.0
 	go.opencensus.io v0.22.4
 	golang.org/x/crypto v0.0.0-20191128160524-b544559bb6d1 // indirect
 	golang.org/x/net v0.0.0-20191126235420-ef20fe5d7933 // indirect
diff --git a/tail/tail.go b/tail/tail.go
index 622d5c54..51ee561b 100644
--- a/tail/tail.go
+++ b/tail/tail.go
@@ -207,6 +207,9 @@ func (t *Tailer) Read(b []byte) (int, error) {
 	}
 }
 
+// openSegment finds a WAL segment with a name that parses to n. This
+// way we do not need to know how wide the segment filename is (i.e.,
+// how many zeros to pad with).
 func openSegment(dir string, n int) (io.ReadCloser, error) {
 	files, err := fileutil.ReadDir(dir)
 	if err != nil {
@@ -214,12 +217,9 @@ func openSegment(dir string, n int) (io.ReadCloser, error) {
 	}
 	for _, fn := range files {
 		k, err := strconv.Atoi(fn)
-		if err != nil || k < n {
+		if err != nil || k != n {
 			continue
 		}
-		if k > n {
-			return nil, errors.Errorf("next segment %d too high, expected %d", n, k)
-		}
 		return wal.OpenReadSegment(filepath.Join(dir, fn))
 	}
 	return nil, tsdb.ErrNotFound
diff --git a/tail/tail_test.go b/tail/tail_test.go
index 08b741a1..c8c72e78 100644
--- a/tail/tail_test.go
+++ b/tail/tail_test.go
@@ -17,15 +17,44 @@ package tail
 import (
 	"bytes"
 	"context"
+	"fmt"
 	"io/ioutil"
 	"math/rand"
 	"os"
+	"path"
 	"testing"
 	"time"
 
 	"github.com/prometheus/tsdb/wal"
+	"github.com/stretchr/testify/require"
 )
 
+func TestOpenSegment(t *testing.T) {
+	dir, err := ioutil.TempDir("", "test_open_segment")
+	if err != nil {
+		t.Fatal(err)
+	}
+	defer os.RemoveAll(dir)
+
+	require.NoError(t, ioutil.WriteFile(path.Join(dir, "000000000000000000000nonsense"), []byte("bad"), 0777))
+
+	for i := 0; i < 10; i++ {
+		require.NoError(t, ioutil.WriteFile(path.Join(dir, fmt.Sprint("000000000000000000000", i)), []byte(fmt.Sprint(i)), 0777))
+	}
+	for i := 19; i >= 10; i-- {
+		require.NoError(t, ioutil.WriteFile(path.Join(dir, fmt.Sprint("000000000000000000000", i)), []byte(fmt.Sprint(i)), 0777))
+	}
+
+	for i := 0; i < 20; i++ {
+		rc, err := openSegment(dir, i)
+		require.NoError(t, err)
+		body, err := ioutil.ReadAll(rc)
+		require.NoError(t, err)
+		require.Equal(t, fmt.Sprint(i), string(body))
+		require.NoError(t, rc.Close())
+	}
+}
+
 func TestTailFuzz(t *testing.T) {
 	dir, err := ioutil.TempDir("", "test_tail")
 	if err != nil {