diff --git a/common.go b/common.go
index f8243f1..ed987ac 100644
--- a/common.go
+++ b/common.go
@@ -1394,7 +1394,11 @@ func (chi *ClientHelloInfo) SupportsCertificate(c *Certificate) error {
 	}
 
 	// The only signed key exchange we support is ECDHE.
-	if !supportsECDHE(config, vers, chi.SupportedCurves, chi.SupportedPoints) {
+	ecdheSupported, err := supportsECDHE(config, vers, chi.SupportedCurves, chi.SupportedPoints)
+	if err != nil {
+		return err
+	}
+	if !ecdheSupported {
 		return supportsRSAFallback(errors.New("client doesn't support ECDHE, can only use legacy RSA key exchange"))
 	}
 
diff --git a/handshake_client.go b/handshake_client.go
index 4ad6aa2..de5c420 100644
--- a/handshake_client.go
+++ b/handshake_client.go
@@ -886,6 +886,19 @@ func (hs *clientHandshakeState) processServerHello() (bool, error) {
 		return false, errors.New("tls: server selected unsupported compression format")
 	}
 
+	supportsPointFormat := false
+	offeredNonCompressedFormat := false
+	for _, format := range hs.serverHello.supportedPoints {
+		if format == pointFormatUncompressed {
+			supportsPointFormat = true
+		} else {
+			offeredNonCompressedFormat = true
+		}
+	}
+	if !supportsPointFormat && offeredNonCompressedFormat {
+		return false, errors.New("tls: server offered only incompatible point formats")
+	}
+
 	if c.handshakes == 0 && hs.serverHello.secureRenegotiationSupported {
 		c.secureRenegotiation = true
 		if len(hs.serverHello.secureRenegotiation) != 0 {
diff --git a/handshake_server.go b/handshake_server.go
index 03d0553..3d5ea81 100644
--- a/handshake_server.go
+++ b/handshake_server.go
@@ -268,7 +268,11 @@ func (hs *serverHandshakeState) processClientHello() error {
 		hs.hello.scts = hs.cert.SignedCertificateTimestamps
 	}
 
-	hs.ecdheOk = supportsECDHE(c.config, c.vers, hs.clientHello.supportedCurves, hs.clientHello.supportedPoints)
+	hs.ecdheOk, err = supportsECDHE(c.config, c.vers, hs.clientHello.supportedCurves, hs.clientHello.supportedPoints)
+	if err != nil {
+		c.sendAlert(alertMissingExtension)
+		return err
+	}
 
 	if hs.ecdheOk && len(hs.clientHello.supportedPoints) > 0 {
 		// Although omitting the ec_point_formats extension is permitted, some
@@ -339,7 +343,7 @@ func negotiateALPN(serverProtos, clientProtos []string, quic bool) (string, erro
 
 // supportsECDHE returns whether ECDHE key exchanges can be used with this
 // pre-TLS 1.3 client.
-func supportsECDHE(c *Config, version uint16, supportedCurves []CurveID, supportedPoints []uint8) bool {
+func supportsECDHE(c *Config, version uint16, supportedCurves []CurveID, supportedPoints []uint8) (bool, error) {
 	supportsCurve := false
 	for _, curve := range supportedCurves {
 		if c.supportsCurve(version, curve) {
@@ -349,10 +353,12 @@ func supportsECDHE(c *Config, version uint16, supportedCurves []CurveID, support
 	}
 
 	supportsPointFormat := false
+	offeredNonCompressedFormat := false
 	for _, pointFormat := range supportedPoints {
 		if pointFormat == pointFormatUncompressed {
 			supportsPointFormat = true
-			break
+		} else {
+			offeredNonCompressedFormat = true
 		}
 	}
 	// Per RFC 8422, Section 5.1.2, if the Supported Point Formats extension is
@@ -361,9 +367,11 @@ func supportsECDHE(c *Config, version uint16, supportedCurves []CurveID, support
 	// the parser. See https://go.dev/issue/49126.
 	if len(supportedPoints) == 0 {
 		supportsPointFormat = true
+	} else if offeredNonCompressedFormat && !supportsPointFormat {
+		return false, errors.New("tls: client offered only incompatible point formats")
 	}
 
-	return supportsCurve && supportsPointFormat
+	return supportsCurve && supportsPointFormat, nil
 }
 
 func (hs *serverHandshakeState) pickCipherSuite() error {