Background/Objectives: Access delays in specialist consultations and diagnostics are frequently cited as key weaknesses of the Polish healthcare system. This study aimed to identify patient- and healthcare employee-reported bottlenecks beyond primary care, focusing on access, organisational and information barriers. Methods: A cross-sectional survey in Primed Medical Center in Lublin (south-eastern Poland) analysed fifty eligible adult respondents (58% patients; 42% healthcare employees). Measures covered access and organisational barriers (primary care, specialists/diagnostics, out-of-hours), perceived quality, equity, and satisfaction. Results: Overall dissatisfaction predominated (66.0% rather/definitely dissatisfied vs. 24.0% somewhat/definitely satisfied), and 70.0% indicated that reform is needed. The most frequent constraints concerned appointment scheduling convenience (88.0%), limited specialist access (86.0%), inability to obtain timely diagnostics (80.0%), unclear guidance on where to seek help (78.0%), and low administrative efficiency (74.0%). Additional concerns included out-of-hours access (60.0% reported no immediate night help) and perceived inequity (58.0% reported unequal access; 62.0% reported unequal treatment). In contrast, primary care availability was rated positively by 78% on a qualitative scale, and physician competence by 62%. Associations with sex, age, residence, and role were significant but small to moderate. Conclusions: Respondents differentiate clinical competence from system performance: negative assessments cluster around organisational barriers and capacity constraints in specialist and diagnostic pathways. Improving patient navigation and information, scheduling and administrative workflows, and specialist/diagnostic capacity—while strengthening primary care coordination—may reduce delays and support more equitable, higher-quality care.