Specialized MP3 splitters, like mp3DirectCut, cut on frame boundaries. But most MP3s use the bit reservoir to conserve space; a frame's audio data often starts in a prior frame, sometimes 2 or 3 frames back. Thus the frames near the split points are likely unplayable and are silently skipped. If the split occurs in the middle of silence, it's probably of no concern.
If the split occurs in the middle of continuous music, you might notice the skipped frames. To help mitigate this, use mp3packer to expand the mp3 to 320 kbps with minimal bit reservoir usage (
-b 320 -r in.mp3 temp.mp3), make your edits in mp3DirectCut, then use mp3packer again to repack (
-s -t -z temp.mp3 out.mp3). It's not guaranteed to work because bit reservoir usage might be unavoidable in a high-bitrate file, but it can help, and doesn't result in quality loss.
Alternatively, you can use pcutmp3 to do the split. It preserves the necessary frames and adds gapless playback data (encoder delay & padding info) to a LAME tag at the beginning of the file. Players which support this data will trim the extra samples upon playback.